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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)~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:
(f∘g)(x)=x(f\circ{g})(x)=x(f∘g)(x)=x
(g∘f)(x)=x(g\circ{f})(x)=x(g∘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))(f∘g)(x)=f(g(x))(g∘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)f(x) at more than one point, then f(x)f\left(x\right)f(x) 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^2y=x2.

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(x)=y=x2x=y2y=±x​f−1(x)=±x​​    Switch x & y    Solve for y​
f−1(x)=±xf^{-1}(x)=\pm\sqrt{x}f−1(x)=±x​

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

Part 2. 

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


For f−1(x)=±x−12f^{-1}(x)=\pm\sqrt{\frac{x-1}{2}}f−1(x)=±2x−1​​:
        Domain: [1,∞)[1,\infin)[1,∞)
        Range: (−∞,∞)(-\infin,\infin)(−∞,∞)

If f−1(x)=2xx−3,f^{-1}(x)=\frac{2x}{x-3},f−1(x)=x−32x​, then determine f(x)f(x)f(x).

Enter your answer as

f(x)=...

I don't know

Extra Practice

Inverse Functions

Let f(x)=3xx+4\displaystyle f(x)=\frac{3x}{x+4}f(x)=x+43x​. 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)=−8x3x−5,f^{-1}(x)=\dfrac{-8x}{3x-5},f−1(x)=3x−5−8x​, then determine f(x)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,f−1(x)=2x−3x+4​−1, then determine f(x)f(x)f(x).

Inverse Functions

Practice: Inverse Functions
Let f(x)=5−4x+6f(x)=5-\dfrac{4}{x+6}f(x)=5−x+64​. Find f−1(x)f^{-1}(x)f−1(x).

Inverse Functions

Practice: Inverse Functions
Let f(x)=32x+1f(x)=\dfrac{3}{2x+1}f(x)=2x+13​. Find f−1(x)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=05x+8y−13=0 is f−1(x)=−85x+135f^{-1}(x)=-\dfrac{8}{5}x+\dfrac{13}{5}f−1(x)=−58​x+513​

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+2y=−52​(x+3)2+2 is f−1(x)=3±5−5x2f^{-1}(x)=3\pm\sqrt{5-\dfrac{5x}{2}}f−1(x)=3±5−25x​​