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(2.6) Inverse Functions

Consider the following function.

When the temperature of water is decreased, it turns into

The inverse of $f(x)$ is the function that turns the output of $f(x)$ into the input of $f(x)$.

To turn the ice back into water, the temperature of the ice must be

The inverse of $f\left(x\right)$ is denoted $f^{-1}\left(x\right)$.
The diagram illustrates $f^{-1}(f($$))$

For all functions that have inverses, $f^{-1}\left(f\left(x\right)\right)=$

What happens if we reverse the order? The diagram illustrates $f(f^{-1}($$))$

For all functions that have inverses, $f\left(f^{-1}\left(x\right)\right)=$

$f^{-1}\left(x\right)$ as a Mapping Diagram

For this example, $f\left(x\right)=$

For this example, $f^{-1}\left(x\right)=$

Example 1


If $f\left(x\right)=3x$, $f^{-1}\left(x\right)=$

Example 2


If $f\left(x\right)=2x-1$, $f^{-1}\left(x\right)=$

Finding $f^{-1}\left(x\right)$ Algebraically

$f^{-1}\left(x\right)$ can be found by:
Step 1. Rewriting $f\left(x\right)$ as $y$.
Step 2. Interchanging $x$ and $y$.
Step 3. Solving for $y$. This is $f^{-1}\left(x\right)$.

$$\begin{align} \class{Step0}{f\left(x\right) \;}&\class{Step0}{= 2x-1}\\ \class{Step1}{ y \;}&\class{Step1}{= 2x-1}\\ \class{Step2}{ \color{red}{x} \;}&\class{Step2}{= 2\color{red}{y} -1 \;}&&\class{Step2}{ \color{red}{\text{variable interchange}} }\\ \class{Step3}{ x+1 \;}&\class{Step3}{= 2y}\\ \class{Step4}{ \frac{x+1}{2} \;}&\class{Step4}{= y\;}\end{align}$$


so, $f^{-1}\left(x\right)=$

Practice

Find the inverse function, $f^{-1}\left(x\right)$, for:

$f\left(x\right)=4x+5,\quad f^{-1}\left(x\right)=$

$f\left(x\right)=1-\frac{1}{2}x,\quad f^{-1}\left(x\right)=$

$f\left(x\right)=\sqrt{1-x},\quad f^{-1}\left(x\right)=$

$f\left(x\right)=\frac{2}{x+1},\quad f^{-1}\left(x\right)=$