High School
SAT
SAT Elite 1500
SAT Tutoring
ACT
ACT Elite 33
ACT Tutoring
University
MCAT
MCAT Elite 515
Med-School Admissions
Pre-Med Tutoring
Pre-Med Plus
LSAT
LSAT Elite 170
LSAT Self-Paced
LSAT Tutoring
DAT
DAT Elite
DAT Tutoring
Log in
Get Started for Free
Inverse Trigonometric Functions
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Logarithmic Functions
4 Activities
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Trigonometric Functions
5 Activities
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Inverse Trigonometric Functions
4 Activities
Which of the following results in a value that is positive:
a.
ln
(
1
50
)
+
ln
(
51
)
\ln\left(\frac{1}{50}\right)+\ln\left(51\right)
ln
(
50
1
)
+
ln
(
51
)
b.
ln
(
1
π
)
+
ln
(
e
)
\ln\left(\frac{1}{\pi}\right)+\ln\left(e\right)
ln
(
π
1
)
+
ln
(
e
)
c.
log
2
1
8
\log_2\sqrt{\frac{1}{8}}
lo
g
2
8
1
d.
sin
(
5
π
4
)
\sin\left(\frac{5\pi}{4}\right)
sin
(
4
5
π
)
e.
tan
(
4
π
3
)
−
cos
(
11
π
6
)
\tan\left(\frac{4\pi}{3}\right)-\cos\left(\frac{11\pi}{6}\right)
tan
(
3
4
π
)
−
cos
(
6
11
π
)
f.
sin
(
arcsin
(
0.67
)
)
\sin\left(\arcsin\left(0.67\right)\right)
sin
(
arcsin
(
0.67
)
)
g.
tan
(
arctan
(
−
1
3
)
)
\tan\left(\arctan\left(-\frac{1}{\sqrt{3}}\right)\right)
tan
(
arctan
(
−
3
1
)
)
h.
arccos
(
cos
(
−
π
3
)
)
\arccos\left(\cos\left(-\frac{\pi}{3}\right)\right)
arccos
(
cos
(
−
3
π
)
)
I don't know
Check Submission
More Logarithmic Functions Questions:
Find the inverse of the function
y
=
log
5
(
2
x
−
1
)
−
9
y=\log_5\left(2x-1\right)-9
y
=
lo
g
5
(
2
x
−
1
)
−
9
.
Find the inverse of the function
y
=
log
5
(
2
x
−
1
)
−
9
y=\log_5\left(2x-1\right)-9
y
=
lo
g
5
(
2
x
−
1
)
−
9
.
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
x
2
−
2
x
+
1
ln
(
x
2
−
5
x
+
6
)
f(x)=\dfrac{x^2-2x+1}{\sqrt{\ln(x^2-5x+6)}}
f
(
x
)
=
ln
(
x
2
−
5
x
+
6
)
x
2
−
2
x
+
1
Transforming a Log
Given
f
(
x
)
=
log
3
(
2
x
+
4
)
f\left(x\right)=\log_3\left(2x+4\right)
f
(
x
)
=
lo
g
3
(
2
x
+
4
)
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Practice: Solving a logarithmic function
Solve for 𝑥 in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Inverse of Logarithmic Functions
Find the inverse of
y
=
5
ln
(
3
ln
x
−
2
)
y=5\ln\left(3\ln x-2\right)
y
=
5
ln
(
3
ln
x
−
2
)
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Solving exponential functions
Solve for
x
if
32
x
/
3
=
8
x
−
12
32^{x/3} = 8^{x-12}
3
2
x
/3
=
8
x
−
12
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Logarithmic Functions
Find the inverse function of
f
(
x
)
=
2
ln
(
x
+
e
)
f\left(x\right)=2\ln\left(x+e\right)\
f
(
x
)
=
2
ln
(
x
+
e
)
.
Logarithmic Functions
Find the inverse function of
f
(
x
)
=
2
ln
(
x
+
e
)
f\left(x\right)=2\ln\left(x+e\right)\
f
(
x
)
=
2
ln
(
x
+
e
)
.
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Domain of Log Functions
Determine the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Solving exponential functions
Solve for
x
if
32
x
/
3
=
8
x
−
12
32^{x/3} = 8^{x-12}
3
2
x
/3
=
8
x
−
12
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Additional Practice Problems--Functions Part 2
Additional Practice Problems
Simplify the following expressions:
a.
ln
(
3
e
2
)
+
e
2
ln
3
\ln\left(3e^2\right)+e^{2\ln3}
ln
(
3
e
2
)
+
e
2
l
n
3
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Logarithmic Functions: Log Values
Practice: Log Values
If
x
=
log
2
1
8
x=\log_2\sqrt{\frac{1}{8}}
x
=
lo
g
2
8
1
,
y
=
ln
(
10
e
2
)
+
ln
(
0.1
)
y=\ln\left(\frac{10}{e^2}\right)+\ln\left(0.1\right)
y
=
ln
(
e
2
10
)
+
ln
(
0.1
)
, and
z
=
e
−
2
ln
3
z=e^{-2\ln3}
z
=
e
−
2
l
n
3
, which of the following statements is true?
Q
:
\bf{Q:}
Q
:
Find the inverse function of
𝑓
(
𝑥
)
=
2
ln
(
𝑥
+
e
)
𝑓(𝑥) = 2 \ln (𝑥 + e)
f
(
x
)
=
2
ln
(
x
+
e
)
Q
:
\bf{Q:}
Q
:
Find the inverse of
f
(
x
)
=
ln
(
x
−
4
)
f\left(x\right)=\ln\left(x-4\right)
f
(
x
)
=
ln
(
x
−
4
)
and state the domain and range for both
f
f
f
and
f
−
1
f^{-1}
f
−
1
.
Practice: Combining Logs
Q
:
\bf{Q:}
Q
:
Evaluate
log
2
1
16
−
log
2
15
+
log
2
30
\log_2\ \dfrac{1}{16}-\log_215+\log_230
lo
g
2
16
1
−
lo
g
2
15
+
lo
g
2
30
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
1
−
x
log
(
1
+
x
)
f(x)=\dfrac{\sqrt{1-x}}{\log\left(1+x\right)}
f
(
x
)
=
lo
g
(
1
+
x
)
1
−
x
Logarithmic Functions: Determine the domain
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
[
ln
(
x
2
)
]
f\left(x\right)=\ln\left[\ln\left(x^2\right)\right]
f
(
x
)
=
ln
[
ln
(
x
2
)
]
.
Logarithmic Functions
Solve for
x
x
x
in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Logarithmic Functions
Find the inverse of
y
=
5
ln
(
3
ln
x
−
2
)
y=5\ln\left(3\ln x-2\right)
y
=
5
ln
(
3
ln
x
−
2
)
Range of Inverse
Find the range of the inverse of
y
=
x
+
2
+
ln
(
5
−
x
)
y=\sqrt{x+2}+\ln(5-x)
y
=
x
+
2
+
ln
(
5
−
x
)
.
Even or Odd Logarithmic Functions
Determine if the following function is even, odd, or neither:
f
(
x
)
=
log
3
1
−
x
1
+
x
f\left(x\right)=\log^3\dfrac{1-x}{1+x}
f
(
x
)
=
lo
g
3
1
+
x
1
−
x
Q
:
\bf{Q:}
Q
:
State the possible values of
f
f
f
and
g
g
g
given
f
(
g
(
x
)
)
=
x
−
ln
x
f\left(g\left(x\right)\right)=\sqrt{x-\ln x}
f
(
g
(
x
)
)
=
x
−
ln
x
Logarithmic Functions
Find the domain of
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
3
4
f\left(x\right)=\left[\ln\left(x-5\right)-2\right]^{^{\frac{3}{4}}}
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
4
3
Inequalities with Logs
Find all
x
x
x
that satisfy the inequality
ln
(
2
x
−
1
)
>
e
3
\ln\left(2x-1\right)>e^3
ln
(
2
x
−
1
)
>
e
3
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Transforming a Log
Given
f
(
x
)
=
log
3
(
2
x
+
4
)
f\left(x\right)=\log_3\left(2x+4\right)
f
(
x
)
=
lo
g
3
(
2
x
+
4
)
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Rules of Logs
Expand the expression
log
[
x
3
sin
(
x
4
)
cos
2
(
5
x
)
(
10
2
x
)
]
\log\left[\dfrac{x^3\sin\left(x^4\right)}{\cos^2\left(5x\right)\left(10^{2x}\right)}\right]
lo
g
[
cos
2
(
5
x
)
(
1
0
2
x
)
x
3
sin
(
x
4
)
]
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate each logarithm and match it to its corresponding answer.
Logarithmic Functions
If
log
3
=
a
\log{3}=a
lo
g
3
=
a
and
log
5
=
b
\log{5}=b
lo
g
5
=
b
, then determine the following in terms of a and b:
Solving Logarithmic Equations
Solve
log
2
(
2
x
+
1
)
−
log
2
(
x
−
1
)
=
3
\log_2{(2x+1)}-\log_2{(x-1)}=3
lo
g
2
(
2
x
+
1
)
−
lo
g
2
(
x
−
1
)
=
3
.
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If
log
7
343
=
b
−
a
\log_{7}{343}=b-a
lo
g
7
343
=
b
−
a
and
log
6
36
=
2
a
+
2
b
\log_{6}{36}=2a+2b
lo
g
6
36
=
2
a
+
2
b
, what is
a
a
a
and
b
b
b
?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If
log
5
625
=
a
+
b
\log_{5}{625}=a+b
lo
g
5
625
=
a
+
b
and
log
3
27
=
2
a
−
b
\log_{3}{27}=2a-b
lo
g
3
27
=
2
a
−
b
, what is
a
a
a
and
b
b
b
?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log
5
6
+
log
5
2
x
2
=
log
5
48
\log_5{6} + \log_5{2x^2}=\log_5{48}
lo
g
5
6
+
lo
g
5
2
x
2
=
lo
g
5
48
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log
(
x
−
3
)
−
log
(
x
−
5
)
=
log
5
\log(x-3)-\log(x-5)=\log5
lo
g
(
x
−
3
)
−
lo
g
(
x
−
5
)
=
lo
g
5
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log
8
2
+
log
8
4
x
2
=
1
\log_82+\log_84x^2=1
lo
g
8
2
+
lo
g
8
4
x
2
=
1
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log
9
(
x
+
6
)
−
log
9
x
=
log
9
2
\log_9(x+6)-\log_9x=\log_92
lo
g
9
(
x
+
6
)
−
lo
g
9
x
=
lo
g
9
2
Logarithm Laws
Practice: Logarithm Laws
If
log
50
=
9.5
\log50=9.5
lo
g
50
=
9.5
, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If
log
20
=
3.4872
\log20=3.4872
lo
g
20
=
3.4872
, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If
log
5
=
a
\log5=a
lo
g
5
=
a
and
log
7
=
b
\log7=b
lo
g
7
=
b
, write each logarithm in terms of
a
a
a
and
b
b
b
.
Logarithm Laws
Practice: Logarithm Laws
If
log
3
=
a
\log3=a
lo
g
3
=
a
and
log
5
=
b
\log5=b
lo
g
5
=
b
, write each logarithm in terms of
a
a
a
and
b
b
b
.
Logarithm Laws
Practice: Logarithm Laws
Evaluate
log
3
(
81
27
3
4
3
5
3
9
6
5
)
\log_{3}{\Bigg(\dfrac{81\sqrt[4]{27^3}3^{\frac{5}{3}}}{9^{\frac{6}{5}}}\Bigg)}
lo
g
3
(
9
5
6
81
4
2
7
3
3
3
5
)
. Leave answer in exact form.
Logarithm Laws
Practice: Logarithm Laws
Which of the following is equivalent to
log
a
(
a
4
b
3
c
5
d
3
2
e
2
)
\log_{a}{\Bigg(\dfrac{a^4\sqrt[3]{b}c^5}{d^{\frac{3}{2}}e^2}\Bigg)}
lo
g
a
(
d
2
3
e
2
a
4
3
b
c
5
)
?
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate
log
1
16
2
=
128
\log_{\frac{1}{16}}{^2=\sqrt{128}}
lo
g
16
1
2
=
128
.
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate
log
1
9
3
27
\log_{\frac{1}{9}}{^3\sqrt{27}}
lo
g
9
1
3
27
.
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate each logarithm and match it to its corresponding answer.
Basics of Logarithms
Practice: Basics of Logarithms
Match each exponential expression to its inverse.
Basics of Logarithms
Practice: Basics of Logarithms
Match each exponential expression to its inverse.
Question 1
Suppose that you have conducted an experiment resulting in the data plotted below.
The following functions
f
(
x
)
f(x)
f
(
x
)
depend on a parameter
b
>
0
b > 0
b
>
0
. Which function would you use to fit the data points?
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Transforming a Log
Given
f
(
x
)
=
log
3
(
2
x
+
4
)
f\left(x\right)=\log_3\left(2x+4\right)
f
(
x
)
=
lo
g
3
(
2
x
+
4
)
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Find the domain of the function
f
(
x
)
=
8
−
4
ln
(
x
)
f\left(x\right)=\sqrt{8-4\ln\left(x\right)}
f
(
x
)
=
8
−
4
ln
(
x
)
When we plot the function
y
=
10000
x
3
y=10000x^3
y
=
10000
x
3
on a log-log graph, we get
Find the domain of the function
f
(
x
)
=
ln
(
[
[
x
]
]
−
4
)
f\left(x\right)=\ln\left(\left[\left[x\right]\right]-4\right)
f
(
x
)
=
ln
(
[
[
x
]
]
−
4
)
where
[
[
x
]
]
\left[\left[x\right]\right]
[
[
x
]
]
represents the greatest integer of x.
Logarithmic Functions
Find the inverse of the function
f
(
x
)
=
2
−
e
3
x
+
1
f(x) = 2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
and find its domain.
Logarithmic Functions
Find the inverse of
f
(
x
)
=
2
ln
(
x
+
e
)
f(x) = 2\ln(x+e)
f
(
x
)
=
2
ln
(
x
+
e
)
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f(x) = \ln(\ln(x^2))
f
(
x
)
=
ln
(
ln
(
x
2
))
Solving exponential functions
Solve for
x
if
32
x
/
3
=
8
x
−
12
32^{x/3} = 8^{x-12}
3
2
x
/3
=
8
x
−
12
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Practice: Solving a logarithmic function
Practice Question: Solving a Logarithmic Function
Solve for 𝑥 in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Enter your answers as numbers only. If there is more than 1 answer for x, type them from smallest to biggest, separated by a comma with no spaces in between (for example, -5,1,2)
Logarithmic Functions
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
1
2
[
log
3
(
x
−
1
)
+
log
3
(
x
−
3
)
]
\frac{1}{2}\left[\log_3\left(x-1\right)+\log_3\left(x-3\right)\right]
2
1
[
lo
g
3
(
x
−
1
)
+
lo
g
3
(
x
−
3
)
]
Find the inverse of the function
y
=
log
5
(
2
x
−
1
)
−
9
y=\log_5\left(2x-1\right)-9
y
=
lo
g
5
(
2
x
−
1
)
−
9
.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Rules of Logs
Expand the expression
log
[
x
3
sin
(
x
4
)
cos
2
(
5
x
)
(
10
2
x
)
]
\log\left[\dfrac{x^3\sin\left(x^4\right)}{\cos^2\left(5x\right)\left(10^{2x}\right)}\right]
lo
g
[
cos
2
(
5
x
)
(
1
0
2
x
)
x
3
sin
(
x
4
)
]
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Inequalities with Logs
Find all
x
x
x
that satisfy the inequality
ln
(
2
x
−
1
)
>
e
3
\ln\left(2x-1\right)>e^3
ln
(
2
x
−
1
)
>
e
3
Inverse of Logarithmic Functions
Find the inverse of
y
=
5
ln
(
3
ln
x
−
2
)
y=5\ln\left(3\ln x-2\right)
y
=
5
ln
(
3
ln
x
−
2
)
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Even or Odd Logarithmic Functions
Determine if the following function is even, odd, or neither:
f
(
x
)
=
log
3
1
−
x
1
+
x
f\left(x\right)=\log^3\dfrac{1-x}{1+x}
f
(
x
)
=
lo
g
3
1
+
x
1
−
x
Logarithmic Functions
Find the domain of
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
3
4
f\left(x\right)=\left[\ln\left(x-5\right)-2\right]^{^{\frac{3}{4}}}
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
4
3
Logarithmic Functions: Determine the domain
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
[
ln
(
x
2
)
]
f\left(x\right)=\ln\left[\ln\left(x^2\right)\right]
f
(
x
)
=
ln
[
ln
(
x
2
)
]
.
Logarithmic Functions
Solve for
x
x
x
in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
1
−
x
log
(
1
+
x
)
f(x)=\dfrac{\sqrt{1-x}}{\log\left(1+x\right)}
f
(
x
)
=
lo
g
(
1
+
x
)
1
−
x
Q
:
\bf{Q:}
Q
:
Find the inverse function of
𝑓
(
𝑥
)
=
2
ln
(
𝑥
+
e
)
𝑓(𝑥) = 2 \ln (𝑥 + e)
f
(
x
)
=
2
ln
(
x
+
e
)
Range of Inverse
Find the range of the inverse of
y
=
x
+
2
+
ln
(
5
−
x
)
y=\sqrt{x+2}+\ln(5-x)
y
=
x
+
2
+
ln
(
5
−
x
)
.
Q
:
\bf{Q:}
Q
:
State the possible values of
f
f
f
and
g
g
g
given
f
(
g
(
x
)
)
=
x
−
ln
x
f\left(g\left(x\right)\right)=\sqrt{x-\ln x}
f
(
g
(
x
)
)
=
x
−
ln
x
Q
:
\bf{Q:}
Q
:
Find the inverse of
f
(
x
)
=
ln
(
x
−
4
)
f\left(x\right)=\ln\left(x-4\right)
f
(
x
)
=
ln
(
x
−
4
)
and state the domain and range for both
f
f
f
and
f
−
1
f^{-1}
f
−
1
.
Practice: Combining Logs
Q
:
\bf{Q:}
Q
:
Evaluate
log
2
1
16
−
log
2
15
+
log
2
30
\log_2\ \dfrac{1}{16}-\log_215+\log_230
lo
g
2
16
1
−
lo
g
2
15
+
lo
g
2
30
What value of
x
x
x
solves the equation
log
2
(
x
)
+
log
2
(
x
−
2
)
=
3
\log_2\left(x\right)+\log_2\left(x-2\right)=3
lo
g
2
(
x
)
+
lo
g
2
(
x
−
2
)
=
3
?
Logarithmic Functions
Consider the functions
u
(
t
)
=
(
t
+
2
)
2
u(t)=(t+2)^2
u
(
t
)
=
(
t
+
2
)
2
and
v
(
t
)
=
ln
(
t
)
v(t)=\ln(\sqrt t)
v
(
t
)
=
ln
(
t
)
. Find
(
v
∘
u
)
(
t
)
(v \circ u)(t)
(
v
∘
u
)
(
t
)
and the domain.
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
3
−
log
3
(
1
x
2
)
\displaystyle3^{-\log_3\left(\frac{1}{x^2}\right)}
3
−
l
o
g
3
(
x
2
1
)
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
e
ln
(
ln
e
7
)
+
ln
(
e
e
ln
6
)
e^{\ln\left(\ln\ e^7\right)}+\ln\left(e^{e^{\ln6}}\right)
e
l
n
(
l
n
e
7
)
+
ln
(
e
e
l
n
6
)
Logarithmic Functions
Simplify
log
7
8
log
7
32
\frac{\log_78}{\log_732}
l
o
g
7
32
l
o
g
7
8
Logarithmic Functions
Evaluate
e
[
2
ln
4
−
ln
2
]
+
ln
(
e
3
e
2
)
e^{[2\ln4-\ln2]}+\ln(e^3e^2)
e
[
2
l
n
4
−
l
n
2
]
+
ln
(
e
3
e
2
)
.
Logarithmic Functions
Find the inverse of
f
(
x
)
=
ln
(
x
−
1
x
+
1
)
f\left(x\right)=\ln\left(\frac{x-1}{x+1}\right)
f
(
x
)
=
ln
(
x
+
1
x
−
1
)
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
x
2
−
2
x
+
1
ln
(
x
2
−
5
x
+
6
)
f(x)=\dfrac{x^2-2x+1}{\sqrt{\ln(x^2-5x+6)}}
f
(
x
)
=
ln
(
x
2
−
5
x
+
6
)
x
2
−
2
x
+
1
Practice: Inverse function
Find the inverse of
f
(
x
)
=
ln
1
8
−
2
−
3
cos
x
f\left(x\right)=\ln\frac{1}{\sqrt{8-2^{-3\cos x}}}
f
(
x
)
=
ln
8
−
2
−
3
c
o
s
x
1
,
0
≤
x
<
π
0 \leq x < \pi
0
≤
x
<
π
.
Logarithmic Functions
Solve for
x
x
x
in the equation
ln
(
ln
(
x
+
1
)
)
=
0
\ln\left(\ln\left(x+1\right)\right)=0
ln
(
ln
(
x
+
1
)
)
=
0
Logarithmic Functions
Solve for
x
x
x
in the equation
ln
(
3
x
−
1
)
=
−
e
\ln\left(3x-1\right)=-e
ln
(
3
x
−
1
)
=
−
e
.
Logarithmic Functions
Find the inverse function of
f
(
x
)
=
2
ln
(
x
+
e
)
f\left(x\right)=2\ln\left(x+e\right)\
f
(
x
)
=
2
ln
(
x
+
e
)
.
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Solve for
x
x
x
in the equation
ln
(
ln
(
x
+
1
)
)
=
0
\ln\left(\ln\left(x+1\right)\right)=0
ln
(
ln
(
x
+
1
)
)
=
0
Practice: Log & Exponential Function Properties
Practice: Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Practice: Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
a.
f
−
1
(
x
)
=
ln
(
3
x
−
1
)
−
2
f^{-1}\left(x\right)=\ln\left(3x-1\right)-2
f
−
1
(
x
)
=
ln
(
3
x
−
1
)
−
2
; domain of
f
−
1
f^{-1}
f
−
1
is
(
1
3
,
∞
)
\left(\frac{1}{3},\infty\right)
(
3
1
,
∞
)
Practice: Finding an inverse function
Practice Question: Finding an Inverse Function
Find the inverse function of 𝑓(𝑥) = 2ln (𝑥 + 𝑒).
Practice: Domain of Log Functions
Practice: Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Practice: Solving a logarithmic function
Practice Question: Solving a Logarithmic Function
Solve for 𝑥 in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
(
x
2
−
5
)
f\left(x\right)=\ln\left(x^2-5\right)
f
(
x
)
=
ln
(
x
2
−
5
)
.
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Logarithmic Functions: Domain and Range
Find the formula of the function obtained by shifting the graph of
y
=
ln
x
y=\ln x
y
=
ln
x
2 units down, then reflecting the graph about the x-axis, and finally, shifting the graph 1 unit up. State the domain and range of the function.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
b. Simplify
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Logarithmic Functions: Log Values
Practice: Log Values
If
x
=
log
2
1
8
x=\log_2\sqrt{\frac{1}{8}}
x
=
lo
g
2
8
1
,
y
=
ln
(
10
e
2
)
+
ln
(
0.1
)
y=\ln\left(\frac{10}{e^2}\right)+\ln\left(0.1\right)
y
=
ln
(
e
2
10
)
+
ln
(
0.1
)
, and
z
=
e
−
2
ln
3
z=e^{-2\ln3}
z
=
e
−
2
l
n
3
, which of the following statements is true?
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
More Trigonometric Functions Questions:
Trigonometric Functions
Find the number of solutions to the equation cos 2𝑥 = cos 𝑥 − 1 in the interval [0,2𝜋].
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Additional Practice Problems--Functions Part 2
Additional Practice Problems
Simplify the following expressions:
a.
ln
(
3
e
2
)
+
e
2
ln
3
\ln\left(3e^2\right)+e^{2\ln3}
ln
(
3
e
2
)
+
e
2
l
n
3
Trigonometric Functions: Trig Identities
Which of the following equations (if any) is true?
Trigonometric Functions
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
.
Trigonometric Functions
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Practice: Trig Equation
Q
:
\bf{Q:}
Q
:
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Q
:
\bf{Q:}
Q
:
Given
cos
θ
=
12
13
\cos\theta=\dfrac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\dfrac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact value of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Practice: Special Angles
Q
:
\bf{Q:}
Q
:
Evaluate
tan
7
π
6
\tan\dfrac{7\pi}{6}
tan
6
7
π
and
sin
15
π
4
\sin\dfrac{15\pi}{4}
sin
4
15
π
.
Q
:
\bf{Q:}
Q
:
Find all trigonometric function values of the angle
4
π
3
\dfrac{4\pi}{3}
3
4
π
Q
:
\bf{Q:}
Q
:
Find the domain of the following function:
f
(
x
)
=
tan
x
f\left(x\right)=\tan\ x
f
(
x
)
=
tan
x
Trigonometric Functions
Determine the number of solutions the equation
3
cos
x
+
sin
2
x
=
0
\sqrt{3}\cos x+\sin2x=0
3
cos
x
+
sin
2
x
=
0
has in the interval
[
0
,
2
π
]
\left[0,2\pi\right]
[
0
,
2
π
]
.
Even or Odd Trigonometric Functions
Determine if the following functions are even, odd, or neither:
f
(
x
)
=
sin
x
+
cos
x
f\left(x\right)=\sin x+\cos x
f
(
x
)
=
sin
x
+
cos
x
and
g
(
x
)
=
sin
x
sec
x
g\left(x\right)=\dfrac{\sin x}{\sec x}
g
(
x
)
=
sec
x
sin
x
Exponential and Trigonometric Functions
Q
:
\bf{Q:}
Q
:
Find the domain of .
f
(
x
)
=
sin
(
3
x
)
e
4
x
−
1
−
tan
(
2
x
)
f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
f
(
x
)
=
e
4
x
−
1
sin
(
3
x
)
−
tan
(
2
x
)
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Trigonometric Functions
Solve the following equation:
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
2\sin x-2\sqrt{3}\cos x=\sqrt{3}\tan x-3
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
Trigonometric Functions
Given
3
cos
x
+
2
sin
2
x
=
3
3\cos x+2\sin^2x=3
3
cos
x
+
2
sin
2
x
=
3
, solve for
x
x
x
in the domain
(
0
,
−
3
π
2
]
\left(0,-\dfrac{3\pi}{2}\right]
(
0
,
−
2
3
π
]
.
Practice: Transforming a Trig Function
Graph the function:
y
=
3
sin
(
2
x
+
π
2
)
+
1
\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1
y
=
3
sin
(
2
x
+
2
π
)
+
1
Trigonometric Functions
The solutions of the equation
tan
2
x
=
sin
2
x
+
cos
2
x
\tan^2x=\sin^2x+\cos^2x
tan
2
x
=
sin
2
x
+
cos
2
x
in the interval
[
0
,
π
]
\left[0,\pi\right]
[
0
,
π
]
are
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Practice: Transforming a Trig Function
Graph the function:
y
=
3
sin
(
2
x
+
π
2
)
+
1
\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1
y
=
3
sin
(
2
x
+
2
π
)
+
1
Trigonometric Functions
A maintenance technician came to a very expensive hotel to install new shower-heads. The only problem is that ,when turned on, the water from the new shower heads varied over time. When the water was first turned on it was 100 degrees Fahrenheit. After being turned on, the temperature cycled between 80 and 120 degrees Fahrenheit. It took exactly 10 minutes from the time the shower was turned on to cycle back to its original temperature of 100 degrees. Let
T
(
t
)
T(t)
T
(
t
)
measure the temperature of water at time t (in minutes). Which of the following could be an equation for
T
(
t
)
T(t)
T
(
t
)
?
Trigonometric Functions
Find the number of solutions to the equation cos 2𝑥 = cos 𝑥 − 1 in the interval [0,2𝜋].
Trigonometric Functions
Find the number of solutions to the equation
cos
(
2
x
)
=
cos
(
x
)
−
1
\cos\left(2x\right)=\cos\left(x\right)-1
cos
(
2
x
)
=
cos
(
x
)
−
1
in the interval
[
0
,
π
]
\left[0,\pi\right]
[
0
,
π
]
.
Greatest Integer Function
Let
f
(
x
)
=
[
[
sin
x
−
2
]
]
f\left(x\right)=\left[\left[\sin x-2\right]\right]
f
(
x
)
=
[
[
sin
x
−
2
]
]
be the greatest integer of
sin
x
−
2
\sin x-2
sin
x
−
2
on the open interval
(
0
,
π
)
\left(0,\ \pi\right)
(
0
,
π
)
.
The range of
f
(
x
)
f(x)
f
(
x
)
is
Trigonometric Functions
Find all solutions to
sin
(
θ
)
=
sin
(
2
θ
)
\sin(\theta)=\sin(2\theta)
sin
(
θ
)
=
sin
(
2
θ
)
on the interval
[
0
,
2
π
)
[0,2\pi)
[
0
,
2
π
)
Trigonometric Functions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x + \frac{\sin 2x \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Practice: Trig Equation
Practice: Trig Equation
Solve the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
.
Practice: Trig Ratios
Practice: Trig Ratios
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Practice: Trigonometric Modeling
Q
:
\bf{Q:}
Q
:
The temperature of the body oscillates throughout the day and can be modeled by the function:
T
(
t
)
=
37
+
0.4
cos
(
π
12
t
−
π
3
)
T\left(t\right)=37\ +\ 0.4\cos\left(\dfrac{\pi}{12} \ t-\dfrac{\pi}{3}\right)
T
(
t
)
=
37
+
0.4
cos
(
12
π
t
−
3
π
)
where
T
T
T
is the temperature in degrees Celsius, and
t
t
t
is the time in hours after 12pm.
a) What is the maximum and minimum temperatures of the body throughout the day?
b) What is the period of this function? Does this make sense given the context of the problem?
Practice: Trig Equation
Practice: Trig Equation
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Trigonometric Functions
Given
3
cos
x
+
2
sin
2
x
=
3
3\cos x+2\sin^2x=3
3
cos
x
+
2
sin
2
x
=
3
, solve for
x
x
x
in the domain
(
0
,
−
3
π
2
]
\left(0,-\dfrac{3\pi}{2}\right]
(
0
,
−
2
3
π
]
.
Trigonometric Functions
Solve the following equation:
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
2\sin x-2\sqrt{3}\cos x=\sqrt{3}\tan x-3
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
Exponential and Trigonometric Functions
Q
:
\bf{Q:}
Q
:
Find the domain of .
f
(
x
)
=
sin
(
3
x
)
e
4
x
−
1
−
tan
(
2
x
)
f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
f
(
x
)
=
e
4
x
−
1
sin
(
3
x
)
−
tan
(
2
x
)
Q
:
\bf{Q:}
Q
:
Find the domain of the following function:
f
(
x
)
=
tan
x
f\left(x\right)=\tan\ x
f
(
x
)
=
tan
x
Even or Odd Trigonometric Functions
Determine if the following functions are even, odd, or neither:
f
(
x
)
=
sin
x
+
cos
x
f\left(x\right)=\sin x+\cos x
f
(
x
)
=
sin
x
+
cos
x
and
g
(
x
)
=
sin
x
sec
x
g\left(x\right)=\dfrac{\sin x}{\sec x}
g
(
x
)
=
sec
x
sin
x
Q
:
\bf{Q:}
Q
:
Find all trigonometric function values of the angle
4
π
3
\dfrac{4\pi}{3}
3
4
π
Trigonometric Functions
Determine the number of solutions the equation
3
cos
x
+
sin
2
x
=
0
\sqrt{3}\cos x+\sin2x=0
3
cos
x
+
sin
2
x
=
0
has in the interval
[
0
,
2
π
]
\left[0,2\pi\right]
[
0
,
2
π
]
.
Practice: Trig Equation
Q
:
\bf{Q:}
Q
:
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Q
:
\bf{Q:}
Q
:
Given
cos
θ
=
12
13
\cos\theta=\dfrac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\dfrac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact value of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Practice: Special Angles
Q
:
\bf{Q:}
Q
:
Evaluate
tan
7
π
6
\tan\dfrac{7\pi}{6}
tan
6
7
π
and
sin
15
π
4
\sin\dfrac{15\pi}{4}
sin
4
15
π
.
Trigonometric Functions
Calculate the value of the expression
tan
−
1
(
tan
(
4
π
3
)
)
\tan^{-1}\left(\tan\left(\frac{4\pi}{3}\right)\right)
tan
−
1
(
tan
(
3
4
π
)
)
Practice: Trig Equation
Practice: Trig Equation
Solve the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
.
Practice: Trig Ratios
Practice: Trig Ratios
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Trigonometric Functions
Determine the number of solutions the equation
3
cos
(
x
)
+
sin
(
2
x
)
=
0
\sqrt{3}\cos\left(x\right)+\sin\left(2x\right)=0
3
cos
(
x
)
+
sin
(
2
x
)
=
0
has in the interval
[
0
,
2
π
]
\left[0,2\pi\right]
[
0
,
2
π
]
.
Practice: Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
.
Find the number of solutions to the equation
cos
2
x
=
cos
x
−
1
\cos2x=\cos x-1
cos
2
x
=
cos
x
−
1
in the interval [0,2𝜋].
Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
.
Trigonometric Functions
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Trigonometric Functions
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
b. Simplify
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Trigonometric Functions: Trig Identities
Which of the following equations (if any) is true?
Trigonometric Functions
The body temperature
T
(
t
)
T(t)
T
(
t
)
of an individual over a day is a rhythmic process. It reaches a maximum of 38.0 C at 6:00 am and a minimum of 36.2 C at 6:00 pm. Choose the trigonometric function that best describes
T
(
t
)
T(t)
T
(
t
)
, where
t
t
t
is given in hours with
t
=
0
t = 0
t
=
0
taken at midnight (one minute after 11:59 pm, also written as 12:00 am).
More Inverse Trigonometric Functions Questions:
Inverse Trigonometric Functions
Compute the exact value of
arccos
(
cos
(
3
π
4
)
)
\arccos \left ( \cos \left ( \frac{3 \pi}{4} \right ) \right )
arccos
(
cos
(
4
3
π
)
)
Find the exact value of
arcsin
(
sin
(
−
17
π
6
)
)
\arcsin(\sin(-\frac{17\pi}{6}))
arcsin
(
sin
(
−
6
17
π
))
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Domain & Range with Inverse Trig
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f\left(x\right)=\frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Inverse Trigonometric Functions
Evaluate
tan
(
tan
−
1
π
4
)
+
arcsin
(
sin
5
π
4
)
\tan\left(\tan^{-1}\frac{\pi}{4}\right)+\arcsin\left(\sin\frac{5\pi}{4}\right)
tan
(
tan
−
1
4
π
)
+
arcsin
(
sin
4
5
π
)
.
Inverse Trigonometric Functions
Find the exact value of
cos
[
tan
−
1
2
+
tan
−
1
3
]
\cos[\tan^{-1}2+\tan^{-1}3]
cos
[
tan
−
1
2
+
tan
−
1
3
]
.
Inverse Trig with Product
Find the exact value of
sin
[
2
cos
−
1
(
1
5
)
]
\sin\bigg[2\cos^{-1}\bigg(\dfrac{1}{5}\bigg)\bigg]
sin
[
2
cos
−
1
(
5
1
)
]
.
Inverse Trigonometric Functions: Restrict the Domain
Find the exact value of
arccos
(
cos
23
π
3
)
\arccos\bigg(\cos\dfrac{23\pi}{3}\bigg)
arccos
(
cos
3
23
π
)
.
Inverse Trigonometric Functions
Simplify
cos
(
arctan
(
4
)
)
\cos(\arctan(4))
cos
(
arctan
(
4
))
.
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Inverse Trigonometric Functions
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Inverse Trigonometric Functions
Without the using a calculator, compute the exact value of
arctan
(
tan
(
13
π
11
)
)
\arctan \left ( \tan \left ( \frac{13\pi}{11} \right ) \right )
arctan
(
tan
(
11
13
π
)
)
*The
π
\pi
π
symbol is obtained by typing in \pi
Inverse Trigonometric Functions
Answer the following Trig and Inverse Trig questions.
Inverse Trigonometric Functions
Without the use a calculator, compute the exact value of
arccos
(
cos
(
3
π
4
)
)
\arccos \left ( \cos \left ( \frac{3 \pi}{4} \right ) \right )
arccos
(
cos
(
4
3
π
)
)
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Inverse Trigonometric Functions
Find the exact value of
sin
(
2
cos
−
1
(
x
)
)
\sin\left(2\cos^{-1}\left(x\right)\right)
sin
(
2
cos
−
1
(
x
)
)
.
Inverse Trigonometric Functions
Find the value of sin (arccos (− √3/2 )).
Inverse Trigonometric Functions
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Inverse Trigonometric Functions
Simplify the following expression:
sec
2
(
arctan
(
1
/
x
2
)
)
\sec^2(\arctan(1/x^2))
sec
2
(
arctan
(
1/
x
2
))
Inverse Trigonometric Functions
Simplify
sin
(
2
arcsin
x
)
\sin(2\arcsin x)
sin
(
2
arcsin
x
)
Inverse Trigonometric Functions
Find the exact value of
tan
[
arcsin
(
−
3
/
5
)
]
\tan[\arcsin(-3/5)]
tan
[
arcsin
(
−
3/5
)]
Inverse Trigonometric Functions
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin(2\cos^{-1}x)
sin
(
2
cos
−
1
x
)
Inverse Trigonometric Functions
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f(x) = \frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Simplify the following expression:
sin
[
−
tan
−
1
(
x
2
−
1
)
]
\sin[-\tan^{-1}(\sqrt{x^2-1})]
sin
[
−
tan
−
1
(
x
2
−
1
)]
Practice: Trigonometric Modeling
Q
:
\bf{Q:}
Q
:
The temperature of the body oscillates throughout the day and can be modeled by the function:
T
(
t
)
=
37
+
0.4
cos
(
π
12
t
−
π
3
)
T\left(t\right)=37\ +\ 0.4\cos\left(\dfrac{\pi}{12} \ t-\dfrac{\pi}{3}\right)
T
(
t
)
=
37
+
0.4
cos
(
12
π
t
−
3
π
)
where
T
T
T
is the temperature in degrees Celsius, and
t
t
t
is the time in hours after 12pm.
a) What is the maximum and minimum temperatures of the body throughout the day?
b) What is the period of this function? Does this make sense given the context of the problem?
Inverse Trigonometric Functions: Restrict the Domain
Find the exact value of
arccos
(
cos
23
π
3
)
\arccos\bigg(\cos\dfrac{23\pi}{3}\bigg)
arccos
(
cos
3
23
π
)
.
Inverse Trigonometric Functions
Find the exact value of
cos
[
tan
−
1
2
+
tan
−
1
3
]
\cos[\tan^{-1}2+\tan^{-1}3]
cos
[
tan
−
1
2
+
tan
−
1
3
]
.
Inverse Trig with Product
Find the exact value of
sin
[
2
cos
−
1
(
1
5
)
]
\sin\bigg[2\cos^{-1}\bigg(\dfrac{1}{5}\bigg)\bigg]
sin
[
2
cos
−
1
(
5
1
)
]
.
Inverse Trigonometric Functions
Simplify the expression
cos
(
tan
−
1
(
x
2
)
)
\cos\left(\tan^{-1}\left(\frac{x}{2}\right)\right)
cos
(
tan
−
1
(
2
x
)
)
Inverse Trigonometric Functions
Simplify the following expression;
sec
−
1
(
2
)
=
\sec^{-1}(2)=
sec
−
1
(
2
)
=
Inverse Trigonometric Functions
Calculate the value of the expression
sin
−
1
(
sin
(
5
π
3
)
)
\sin^{-1}\left(\sin\left(\frac{5\pi}{3}\right)\right)
sin
−
1
(
sin
(
3
5
π
)
)
Find the exact value of
arcsin
(
sin
(
−
17
π
6
)
)
\arcsin(\sin(-\frac{17\pi}{6}))
arcsin
(
sin
(
−
6
17
π
))
Inverse Trigonometric Functions
Find
sin
(
arctan
(
x
)
)
\sin\left(\arctan\left(x\right)\right)
sin
(
arctan
(
x
)
)
.
Practice: Trig & Inverse Trig Values (level 2)
Practice: Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Domain & Range with Inverse Trig
Practice: Domain & Range with Inverse Trig
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f\left(x\right)=\frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
.
Find the value of
sin
[
arccos
(
−
3
2
)
]
\sin\left[\arccos\left(-\frac{\sqrt{3}}{2}\right)\right]
sin
[
arccos
(
−
2
3
)
]
.
Inverse Trigonometric Functions
Find sin (arctan 𝑥).
Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Domain & Range with Inverse Trig
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f\left(x\right)=\frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
.
Inverse trigonometric functions
Evaluate the expression
arcsin
(
−
1
2
)
×
cos
(
7
π
6
)
+
2
ln
(
3
e
)
\arcsin\left(-\frac{1}{\sqrt{2}}\right)\times\cos\left(\frac{7\pi}{6}\right)+2\ln\left(3\sqrt{e}\right)
arcsin
(
−
2
1
)
×
cos
(
6
7
π
)
+
2
ln
(
3
e
)
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Inverse Trigonometric Functions
Evaluate
tan
(
tan
−
1
π
4
)
+
arcsin
(
sin
5
π
4
)
\tan\left(\tan^{-1}\frac{\pi}{4}\right)+\arcsin\left(\sin\frac{5\pi}{4}\right)
tan
(
tan
−
1
4
π
)
+
arcsin
(
sin
4
5
π
)
.
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.