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Inverse Functions
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Inverse Functions
3 Activities
Which of the following functions have an inverse?
i. 𝑓(𝑥) = 3𝑥 − 1
ii. 𝑓(𝑥) = −5
iii. 𝑓(𝑥) = 2𝑥(𝑥
2
− 1)
iv. 𝑓(𝑥) = −𝑥(𝑥 − 4)
i and ii
i only
ii only
i and iii
none of them
I don't know
Check Submission
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
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
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.