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Properties of Functions
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Exponential Functions
3 Activities
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Inverse Functions
3 Activities
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Even and Odd Functions
2 Activities
Which of the following statements is/are true about the functions
f
(
x
)
=
0
f\left(x\right)=0
f
(
x
)
=
0
,
g
(
x
)
=
x
3
cos
x
g\left(x\right)=x^3\ \cos x
g
(
x
)
=
x
3
cos
x
, and
h
(
x
)
=
e
sin
x
h\left(x\right)=e^{\sin x}
h
(
x
)
=
e
s
i
n
x
?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
iii. Exactly one of the functions has an inverse.
a) i only
b) ii only
c) ii and iii only
d) iii only
e) None of the above
I don't know
Check Submission
More Exponential Functions Questions:
Transforming an Exponential
Given
f
(
x
)
=
−
2
e
3
x
−
6
+
1
f\left(x\right)=-2e^{3x-6}+1
f
(
x
)
=
−
2
e
3
x
−
6
+
1
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Which of the following functions, if any, has an inverse?
i.
f
(
x
)
=
3
x
−
1
f\left(x\right)=3x-1
f
(
x
)
=
3
x
−
1
ii.
f
(
x
)
=
−
5
f\left(x\right)=-5
f
(
x
)
=
−
5
Exponential Functions
Solve for
x
x
x
in the equation
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
3x^2−15=(x^2−5)e^{4-x}
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
Exponential Functions
If
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
3x^2−15=(x^2−5)e^{4-x}
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
, find
x
x
x
Exponential Functions
Solve for
x
x
x
in the equation
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
3x^2−15=(x^2−5)e^{4-x}
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
Properties of Functions
Which of the following statements is/are true about the functions
f
(
x
)
=
0
f\left(x\right)=0
f
(
x
)
=
0
,
g
(
x
)
=
x
3
cos
x
g\left(x\right)=x^3\ \cos x
g
(
x
)
=
x
3
cos
x
, and
h
(
x
)
=
e
sin
x
h\left(x\right)=e^{\sin x}
h
(
x
)
=
e
s
i
n
x
?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Properties of Exponential Functions
If
f
(
x
)
=
−
4
⋅
e
3
x
π
⋅
2
2
x
f\left(x\right)=\frac{-4\cdot e^{3x}}{\pi\cdot2^{2x}}
f
(
x
)
=
π
⋅
2
2
x
−
4
⋅
e
3
x
, which of the following statements is true?
Exponential Functions: Domain & Range
Find the domain and range of
f
(
x
)
=
2
−
9
x
−
8
f\left(x\right)=\sqrt{2^{-9x}-8}
f
(
x
)
=
2
−
9
x
−
8
.
Exponential Functions
Which of the following statements is/are true about the exponential function
f
(
x
)
=
C
4
x
+
b
f\left(x\right)=C4^x+b
f
(
x
)
=
C
4
x
+
b
whose graph passes through the points
(
1
,
1
)
\left(1,1\right)
(
1
,
1
)
and
(
1
2
,
2
)
\left(\frac{1}{2},2\right)
(
2
1
,
2
)
?
i.
f
(
x
)
=
2
(
4
x
)
−
1
f\left(x\right)=2\left(4^x\right)-1
f
(
x
)
=
2
(
4
x
)
−
1
ii. It has a horizontal asymptote at
y
=
−
1
y=-1
y
=
−
1
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to
y
=
e
x
y=e^x
y
=
e
x
:
Shifting the graph 2 units down and 1 unit left
Reflecting along the
x
x
x
axis
Practice: Exponents Equation
Q
:
\bf{Q:}
Q
:
Solve for
x
x
x
in
2
x
2
−
x
=
64
2^{x^2-x}=64
2
x
2
−
x
=
64
Exponential Functions
Find the domain of the function
f
(
x
)
=
1
(
e
x
−
1
)
(
x
+
1
)
+
1
2
−
x
2
f\left(x\right)=\dfrac{1}{\left(e^x-1\right)\left(x+1\right)}+\dfrac{1}{\sqrt{2-x^2}}
f
(
x
)
=
(
e
x
−
1
)
(
x
+
1
)
1
+
2
−
x
2
1
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
)
Exponential Functions: Inequalities with Exponents
Find all
x
x
x
that satisfy the inequality
e
x
>
e
−
x
e^x>e^{−x}
e
x
>
e
−
x
Transforming an Exponential
Given
f
(
x
)
=
−
2
e
3
x
−
6
+
1
f\left(x\right)=-2e^{3x-6}+1
f
(
x
)
=
−
2
e
3
x
−
6
+
1
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Exponential Functions
Solve for
x
x
x
in the equation
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
3x^2−15=(x^2−5)e^{4-x}
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
Solving Exponential Equations
Solve
5
(
1
128
)
2
x
+
5
=
10
(
32
)
x
+
1
5\Bigg(\dfrac{1}{128}\Bigg)^{2x+5}=10(32)^{x+1}
5
(
128
1
)
2
x
+
5
=
10
(
32
)
x
+
1
Simplifying Exponential Expressions
Simplify
(
72
a
b
(
2
a
+
b
)
3
2
a
−
3
b
)
\Bigg(\dfrac{72^{ab}(2^{a+b})}{3^{2a-3b}}\Bigg)
(
3
2
a
−
3
b
7
2
ab
(
2
a
+
b
)
)
Solving Exponential Equations
Solve
4
(
8
)
3
x
+
1
=
32
(
4
)
x
+
2
4(8)^{3x+1}=32(4)^{x+2}
4
(
8
)
3
x
+
1
=
32
(
4
)
x
+
2
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x:
4
x
−
6
(
2
x
)
+
8
=
0
4^x-6(2^x)+8=0
4
x
−
6
(
2
x
)
+
8
=
0
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x:
3
2
x
−
3
x
=
0
3^{2x}-3^x=0
3
2
x
−
3
x
=
0
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of
x
x
x
makes the following statement true? Leave answer in exact form.
(
216
125
)
3
x
=
(
36
25
)
2
x
−
1
\Bigg(\dfrac{216}{125}\Bigg)^{3x}=\Bigg(\dfrac{36}{25}\Bigg)^{2x-1}
(
125
216
)
3
x
=
(
25
36
)
2
x
−
1
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of
x
x
x
makes the following statement true? Leave answer in exact form.
(
3
4
)
2
x
−
1
=
(
81
256
)
x
+
2
\Bigg(\dfrac{3}{4}\Bigg)^{2x-1}=\Bigg(\dfrac{81}{256}\Bigg)^{x+2}
(
4
3
)
2
x
−
1
=
(
256
81
)
x
+
2
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x:
25
4
x
+
5
=
125
x
−
3
25^{4x+5}=125^{x-3}
2
5
4
x
+
5
=
12
5
x
−
3
. Leave answer as a fraction.
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x:
4
2
x
+
1
=
8
3
x
−
1
4^{2x+1}=8^{3x-1}
4
2
x
+
1
=
8
3
x
−
1
. Leave answer as a fraction.
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to
y
=
e
x
y=e^x
y
=
e
x
:
Shifting the graph 2 units down and 1 unit left
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to
y
=
e
x
y=e^x
y
=
e
x
:
Shifting the graph 2 units down and 1 unit left
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Exponential Functions: Domain & Range
Find the domain and range of
f
(
x
)
=
2
−
9
x
−
8
f\left(x\right)=\sqrt{2^{-9x}-8}
f
(
x
)
=
2
−
9
x
−
8
.
Exponential Functions
Find the domain of the function
f
(
x
)
=
e
2
x
−
8
f\left(x\right)=e^{\sqrt{2x-8}}
f
(
x
)
=
e
2
x
−
8
Transformations
Which of the following is the equation of the function produced by vertically shifting the graph of
y
=
e
x
y=e^x
y
=
e
x
up 3 units and then vertically stretching it by a factor of 4?
Q
:
\bf{Q:}
Q
:
The lifespan of a mammal species is related to its heart rate by the function
L
(
h
)
=
120
h
−
0.01
L\left(h\right)=120h^{-0.01}
L
(
h
)
=
120
h
−
0.01
where
L
L
L
is the lifespan in years and
h
h
h
is the heart rate in beats per minute. Which of the following observations are true? Select all that apply.
Note: This question is for practice purposes only, it is not based on real scientific data.
Which of this functions represents a quantity that doubles every
3
3
3
years?
Practice: Solving Exponential Equations
Find all solutions to the equation
4
2
x
⋅
8
x
2
+
x
=
1
4
4^{2x}\cdot8^{x^2+x}=\frac{1}{4}
4
2
x
⋅
8
x
2
+
x
=
4
1
.
Transforming an Exponential
Given
f
(
x
)
=
−
2
e
3
x
−
6
+
1
f\left(x\right)=-2e^{3x-6}+1
f
(
x
)
=
−
2
e
3
x
−
6
+
1
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Exponential Functions
Solve for
x
x
x
in the equation
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
3x^2−15=(x^2−5)e^{4-x}
3
x
2
−
15
=
(
x
2
−
5
)
e
4
−
x
Exponential Functions: Inequalities with Exponents
Find all
x
x
x
that satisfy the inequality
e
x
>
e
−
x
e^x>e^{−x}
e
x
>
e
−
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
)
Exponential Functions
Find the domain of the function
f
(
x
)
=
1
(
e
x
−
1
)
(
x
+
1
)
+
1
2
−
x
2
f\left(x\right)=\dfrac{1}{\left(e^x-1\right)\left(x+1\right)}+\dfrac{1}{\sqrt{2-x^2}}
f
(
x
)
=
(
e
x
−
1
)
(
x
+
1
)
1
+
2
−
x
2
1
Practice: Exponents Equation
Q
:
\bf{Q:}
Q
:
Solve for
x
x
x
in
2
x
2
−
x
=
64
2^{x^2-x}=64
2
x
2
−
x
=
64
Practice: Exponential function
The exponential function
y
=
b
x
+
k
,
(
b
>
0
)
y=b^x+k,\ \left(b>0\right)
y
=
b
x
+
k
,
(
b
>
0
)
passes through the point
(
2
,
13
16
)
\left(2,\frac{13}{16}\right)
(
2
,
16
13
)
and has a horizontal asymptote at
y
=
1
4
y=\frac{1}{4}
y
=
4
1
.
Find the equation of this graph.
Exponential Functions
Solve the equation for
x
x
x
.
e
2
x
=
11
e
x
−
30
e^{2x}=11e^x-30
e
2
x
=
11
e
x
−
30
Exponential Functions
If
h
(
x
)
=
e
x
+
2
x
+
1
h\left(x\right)=e^x+2x+1
h
(
x
)
=
e
x
+
2
x
+
1
, find the value of
h
−
1
(
2
)
h^{-1}\left(2\right)
h
−
1
(
2
)
Exponential Functions
Find the domain of the function
f
(
x
)
=
1
8
−
2
−
3
x
f\left(x\right)=\frac{1}{\sqrt{8-2^{-3x}}}
f
(
x
)
=
8
−
2
−
3
x
1
Practice: Log & Exponential Function Properties
Practice: Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Exponential Functions
The exponential function
y
=
b
x
+
k
,
(
b
>
0
)
y=b^x+k,\ \left(b>0\right)
y
=
b
x
+
k
,
(
b
>
0
)
passes through the point
(
2
,
13
16
)
\left(2,\frac{13}{16}\right)
(
2
,
16
13
)
and has a horizontal asymptote at
y
=
1
4
y=\frac{1}{4}
y
=
4
1
.
Find the equation of this graph.
Which of the following functions, if any, has an inverse?
i.
f
(
x
)
=
3
x
−
1
f\left(x\right)=3x-1
f
(
x
)
=
3
x
−
1
ii.
f
(
x
)
=
−
5
f\left(x\right)=-5
f
(
x
)
=
−
5
Exponential Functions
Find the domain of the function
f
(
x
)
=
1
8
−
2
−
3
x
f\left(x\right)=\frac{1}{\sqrt{8-2^{-3x}}}
f
(
x
)
=
8
−
2
−
3
x
1
Properties of Exponential Functions
If
f
(
x
)
=
−
4
⋅
e
3
x
π
⋅
2
2
x
f\left(x\right)=\frac{-4\cdot e^{3x}}{\pi\cdot2^{2x}}
f
(
x
)
=
π
⋅
2
2
x
−
4
⋅
e
3
x
, which of the following statements is true?
Exponential Functions
Which of the following statements is/are true about the exponential function
f
(
x
)
=
C
4
x
+
b
f\left(x\right)=C4^x+b
f
(
x
)
=
C
4
x
+
b
whose graph passes through the points
(
1
,
1
)
\left(1,1\right)
(
1
,
1
)
and
(
1
2
,
2
)
\left(\frac{1}{2},2\right)
(
2
1
,
2
)
?
i.
f
(
x
)
=
2
(
4
x
)
−
1
f\left(x\right)=2\left(4^x\right)-1
f
(
x
)
=
2
(
4
x
)
−
1
ii. It has a horizontal asymptote at
y
=
−
1
y=-1
y
=
−
1
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to
y
=
e
x
y=e^x
y
=
e
x
:
Shifting the graph 2 units down and 1 unit left
Reflecting along the
x
x
x
axis
More Inverse Functions Questions:
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of
f
(
x
)
=
x
+
3
2
−
5
x
f\left(x\right)=\frac{x+3}{2-5x}
f
(
x
)
=
2
−
5
x
x
+
3
.
Finding the Inverse of a Function
Find the domain of the function
f
(
x
)
=
x
+
2
x
−
1
f\left(x\right)=\frac{x+2}{x-1}
f
(
x
)
=
x
−
1
x
+
2
, and a formula for its inverse.
Inverse Function Values
If
f
(
−
1
2
)
=
e
f\left(-\frac{1}{2}\right)=e
f
(
−
2
1
)
=
e
, and
f
f
f
is a one-to-one function, then we know that
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.
Which of the following functions, if any, has an inverse?
i.
f
(
x
)
=
3
x
−
1
f\left(x\right)=3x-1
f
(
x
)
=
3
x
−
1
ii.
f
(
x
)
=
−
5
f\left(x\right)=-5
f
(
x
)
=
−
5
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
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.
Inverse Functions
Additional Practice Problems
For each of the following functions, find its inverse and state the domain and range:
a.
f
(
x
)
=
−
2
(
x
−
1
)
2
+
3
,
(
x
≥
1
)
f\left(x\right)=-2\left(x-1\right)^2+3,\ \ \ \left(x\ge1\right)
f
(
x
)
=
−
2
(
x
−
1
)
2
+
3
,
(
x
≥
1
)
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.
Finding an inverse function
Find the inverse function of 𝑓(𝑥) = 2ln (𝑥 + 𝑒).
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of
f
(
x
)
=
x
+
3
2
−
5
x
f\left(x\right)=\frac{x+3}{2-5x}
f
(
x
)
=
2
−
5
x
x
+
3
.
Properties of Functions
Which of the following statements is/are true about the functions
f
(
x
)
=
0
f\left(x\right)=0
f
(
x
)
=
0
,
g
(
x
)
=
x
3
cos
x
g\left(x\right)=x^3\ \cos x
g
(
x
)
=
x
3
cos
x
, and
h
(
x
)
=
e
sin
x
h\left(x\right)=e^{\sin x}
h
(
x
)
=
e
s
i
n
x
?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Finding the Inverse of a Function
Find the domain of the function
f
(
x
)
=
x
+
2
x
−
1
f\left(x\right)=\frac{x+2}{x-1}
f
(
x
)
=
x
−
1
x
+
2
, and a formula for its inverse.
Inverse Function Values
If
f
(
−
1
2
)
=
e
f\left(-\frac{1}{2}\right)=e
f
(
−
2
1
)
=
e
, and
f
f
f
is a one-to-one function, then we know that
Inverse Functions
Let
f
(
x
)
=
3
x
x
+
4
\displaystyle f(x)=\frac{3x}{x+4}
f
(
x
)
=
x
+
4
3
x
. Determine
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
.
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let
y
=
f
(
x
)
y=f(x)~
y
=
f
(
x
)
be graphed below:
Find and graph
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let
y
=
f
(
x
)
y=f(x)~
y
=
f
(
x
)
be graphed below:
Find and graph
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
Inverse Functions
Practice: Inverse Functions
If
f
−
1
(
x
)
=
−
8
x
3
x
−
5
,
f^{-1}(x)=\dfrac{-8x}{3x-5},
f
−
1
(
x
)
=
3
x
−
5
−
8
x
,
then determine
f
(
x
)
f(x)
f
(
x
)
.
Inverse Functions
Practice: Inverse Functions
If
f
−
1
(
x
)
=
x
+
4
2
x
−
3
−
1
,
f^{-1}(x)=\dfrac{x+4}{2x-3}-1,
f
−
1
(
x
)
=
2
x
−
3
x
+
4
−
1
,
then determine
f
(
x
)
f(x)
f
(
x
)
.
Inverse Functions
Practice: Inverse Functions
Let
f
(
x
)
=
5
−
4
x
+
6
f(x)=5-\dfrac{4}{x+6}
f
(
x
)
=
5
−
x
+
6
4
. Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
.
Inverse Functions
Practice: Inverse Functions
Let
f
(
x
)
=
3
2
x
+
1
f(x)=\dfrac{3}{2x+1}
f
(
x
)
=
2
x
+
1
3
. Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
.
Inverse Functions
Practice: Inverse Functions
True or False:
The inverse of the function
5
x
+
8
y
−
13
=
0
5x+8y-13=0
5
x
+
8
y
−
13
=
0
is
f
−
1
(
x
)
=
−
8
5
x
+
13
5
f^{-1}(x)=-\dfrac{8}{5}x+\dfrac{13}{5}
f
−
1
(
x
)
=
−
5
8
x
+
5
13
Inverse Functions
Practice: Inverse Functions
True or False:
The inverse of the function
y
=
−
2
5
(
x
+
3
)
2
+
2
y=-\frac{2}{5}\left(x+3\right)^2+2
y
=
−
5
2
(
x
+
3
)
2
+
2
is
f
−
1
(
x
)
=
3
±
5
−
5
x
2
f^{-1}(x)=3\pm\sqrt{5-\dfrac{5x}{2}}
f
−
1
(
x
)
=
3
±
5
−
2
5
x
Inverse Functions
Which of the following functions have an inverse?
i. 𝑓(𝑥) = 3𝑥 − 1
ii. 𝑓(𝑥) = −5
Inverse Functions
Find the inverse of the function
f
(
x
)
=
x
−
1
3
−
2
x
f\left(x\right)=\frac{x-1}{3-2x}
f
(
x
)
=
3
−
2
x
x
−
1
.
Practice: Inverse function
Find the inverse of
y
=
x
+
1
+
2
y=\sqrt{x+1}+2
y
=
x
+
1
+
2
and its domain.
Which of the following functions, if any, has an inverse?
i.
f
(
x
)
=
3
x
−
1
f\left(x\right)=3x-1
f
(
x
)
=
3
x
−
1
ii.
f
(
x
)
=
−
5
f\left(x\right)=-5
f
(
x
)
=
−
5
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
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.
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of
f
(
x
)
=
x
+
3
2
−
5
x
f\left(x\right)=\frac{x+3}{2-5x}
f
(
x
)
=
2
−
5
x
x
+
3
.
Finding an inverse function
Find the inverse function of 𝑓(𝑥) = 2ln (𝑥 + 𝑒).
Inverse Functions
Additional Practice Problems
For each of the following functions, find its inverse and state the domain and range:
a.
f
(
x
)
=
−
2
(
x
−
1
)
2
+
3
,
(
x
≥
1
)
f\left(x\right)=-2\left(x-1\right)^2+3,\ \ \ \left(x\ge1\right)
f
(
x
)
=
−
2
(
x
−
1
)
2
+
3
,
(
x
≥
1
)
Inverse Function Values
If
f
(
−
1
2
)
=
e
f\left(-\frac{1}{2}\right)=e
f
(
−
2
1
)
=
e
, and
f
f
f
is a one-to-one function, then we know that
Finding the Inverse of a Function
Find the domain of the function
f
(
x
)
=
x
+
2
x
−
1
f\left(x\right)=\frac{x+2}{x-1}
f
(
x
)
=
x
−
1
x
+
2
, and a formula for its inverse.
More Even and Odd Functions Questions:
Properties of Functions
Which of the following statements is/are true about the functions
f
(
x
)
=
0
f\left(x\right)=0
f
(
x
)
=
0
,
g
(
x
)
=
x
3
cos
x
g\left(x\right)=x^3\ \cos x
g
(
x
)
=
x
3
cos
x
, and
h
(
x
)
=
e
sin
x
h\left(x\right)=e^{\sin x}
h
(
x
)
=
e
s
i
n
x
?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Combining Families of Functions
Practice: Combining Families of Functions
Is the function
f
(
x
)
=
x
2
x
2
−
6
f(x)=\dfrac{x^2}{x^2-6}
f
(
x
)
=
x
2
−
6
x
2
even, odd, or neither?
Combining Families of Functions
Practice: Combining Families of Functions
Is the function
f
(
x
)
=
x
x
2
−
1
f(x)=x\sqrt{x^2-1}
f
(
x
)
=
x
x
2
−
1
even, odd, or neither?
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function
y
=
−
2
x
3
+
5
y=-2x^3+5
y
=
−
2
x
3
+
5
is odd.
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function
y
=
x
x
2
−
1
y=\dfrac{x}{x^2-1}
y
=
x
2
−
1
x
is odd.