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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Inverse Functions

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Inverse Functions

An inverse function is a function in which the x and y values have switched.
  • Denoted by f−1(x) f^{-1}(x)~
  • Pronounced 'f inverse x'
Two functions, f(x) and g(x), are inverses of each other if the following 2 properties are true:
  1. (f∘g)(x)=x(f\circ{g})(x)=x
  2. (g∘f)(x)=x(g\circ{f})(x)=x

Note: (f∘g)(x)=f(g(x))(g∘f)(x)=g(f(x))(f\circ{g})(x)=f(g(x))\newline{}\newline(g\circ{f})(x)=g(f(x))




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Horizontal Line Test


If any horizontal line intersects the graph of a function f(x)f\left(x\right) at more than one point, then f(x)f\left(x\right) is not one-to-one. If a function is not one-to-one, it cannot have an inverse function.

Example
Find the inverse of the function y=x2y=x^2.
f(x)=y=x2x=y2    Switch x & yy=±x    Solve for yf−1(x)=±x\begin{array} {} f(x)=y=x^2\\\\ x=y^2&~~~~\color{green}\footnotesize{\text{Switch x \& y}}\\\\ y=\pm\sqrt{x}&~~~~\color{green}\footnotesize{\text{Solve for y}}\\\\ f^{-1}(x)=\pm\sqrt{x} \end{array}
f−1(x)=±xf^{-1}(x)=\pm\sqrt{x}

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Example: Inverse Functions

Given f(x)=2x2+1f(x)=2x^2+1:
  1. Find the inverse f−1(x)f^{-1}(x)
  2. Determine the domain and range for f(x)f(x) and compare it to the domain and range for f−1(x)f^{-1}(x)

Part 1.

f(x)=2x2+1x=2f−1(x)2+1x−1=2f−1(x)2f−1(x)=±x−12f(x)=2x^2+1\newline{}\newline x=2{f^{-1}(x)}^2+1\newline{}\newline{} x-1=2{f^{-1}(x)}^2\newline{}\newline f^{-1}(x)=\pm\sqrt{\frac{x-1}{2}}

Part 2.

For f(x)=2x2+1f(x)=2x^2+1:
Domain: (−∞,∞)(-\infin,\infin)
Range: [1,∞)[1,\infin)


For f−1(x)=±x−12f^{-1}(x)=\pm\sqrt{\frac{x-1}{2}}:
Domain: [1,∞)[1,\infin)
Range: (−∞,∞)(-\infin,\infin)
If f−1(x)=2xx−3,f^{-1}(x)=\frac{2x}{x-3}, then determine f(x)f(x).
Extra Practice
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of f(x)=x+32−5xf\left(x\right)=\frac{x+3}{2-5x}.
Finding the Inverse of a Function
Find the domain of the function f(x)=x+2x−1f\left(x\right)=\frac{x+2}{x-1}, and a formula for its inverse.
Inverse Function Values
If f(−12)=ef\left(-\frac{1}{2}\right)=e, and ff 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−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Which of the following functions, if any, has an inverse?
i. f(x)=3x−1f\left(x\right)=3x-1
ii. f(x)=−5f\left(x\right)=-5
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+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)
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+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+32−5xf\left(x\right)=\frac{x+3}{2-5x}.
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin 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+2x−1f\left(x\right)=\frac{x+2}{x-1}, and a formula for its inverse.
Inverse Function Values
If f(−12)=ef\left(-\frac{1}{2}\right)=e, and ff is a one-to-one function, then we know that
Inverse Functions
Let f(x)=3xx+4\displaystyle f(x)=\frac{3x}{x+4}. Determine f−1(x)f^{-1}(x).
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let y=f(x) y=f(x)~ be graphed below:
Find and graph f−1(x)f^{-1}(x)
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let y=f(x) y=f(x)~ be graphed below:
Find and graph f−1(x)f^{-1}(x)
Inverse Functions
Practice: Inverse Functions
If f−1(x)=−8x3x−5,f^{-1}(x)=\dfrac{-8x}{3x-5}, then determine f(x)f(x).
Inverse Functions
Practice: Inverse Functions
If f−1(x)=x+42x−3−1,f^{-1}(x)=\dfrac{x+4}{2x-3}-1, then determine f(x)f(x).
Inverse Functions
Practice: Inverse Functions
Let f(x)=5−4x+6f(x)=5-\dfrac{4}{x+6}. Find f−1(x)f^{-1}(x).
Inverse Functions
Practice: Inverse Functions
Let f(x)=32x+1f(x)=\dfrac{3}{2x+1}. Find f−1(x)f^{-1}(x).
Inverse Functions
Practice: Inverse Functions
True or False:
The inverse of the function 5x+8y−13=05x+8y-13=0 is f−1(x)=−85x+135f^{-1}(x)=-\dfrac{8}{5}x+\dfrac{13}{5}
Inverse Functions
Practice: Inverse Functions
True or False:
The inverse of the function y=−25(x+3)2+2y=-\frac{2}{5}\left(x+3\right)^2+2 is f−1(x)=3±5−5x2f^{-1}(x)=3\pm\sqrt{5-\dfrac{5x}{2}}
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−13−2xf\left(x\right)=\frac{x-1}{3-2x}.
Practice: Inverse function
Find the inverse of y=x+1+2y=\sqrt{x+1}+2 and its domain.
Which of the following functions, if any, has an inverse?
i. f(x)=3x−1f\left(x\right)=3x-1
ii. f(x)=−5f\left(x\right)=-5
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of f(x)=x+32−5xf\left(x\right)=\frac{x+3}{2-5x}.
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)
Inverse Function Values
If f(−12)=ef\left(-\frac{1}{2}\right)=e, and ff 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)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin 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+2x−1f\left(x\right)=\frac{x+2}{x-1}, and a formula for its inverse.