(2.6) Inverse Functions
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, $f^{-1}\left(x\right)=$
If $f\left(x\right)=3x$, $f^{-1}\left(x\right)=$
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 $f^{-1}\left(x\right)$.
$f\left(x\right) = 2x-1$
$y = 2x-1$
${\color{red}x} = 2{\color{red}y} -1 $
$x+1 = 2y$
$\frac{x+1}{2} = y$
$y = 2x-1$
${\color{red}x} = 2{\color{red}y} -1 $
$x+1 = 2y$
$\frac{x+1}{2} = y$
$$\begin{align} \class{Step0}{\;}\end{align}$$
so, $f^{-1}\left(x\right)=$
Practice
Find the inverse function, $f^{-1}\left(x\right)$, for:
$f\left(x\right)=4x+5,\quad f^{-1}\left(x\right)=$
$f\left(x\right)=1-\frac{1}{2}x,\quad f^{-1}\left(x\right)=$
$f\left(x\right)=\sqrt{1-x},\quad f^{-1}\left(x\right)=$
$f\left(x\right)=\frac{2}{x+1},\quad f^{-1}\left(x\right)=$