Q 51. - Precalculus Enhanced with Graphing Utilities

Question

The function $f\left(x\right)=4x+2$ is one-to-one. Find its inverse and check the answer. Graph $f,f^{-1}$ and $y=x$ on the same coordinate axes. 

Step-by-Step Solution

Answer

Inverse of the function $f\left(x\right)=4x+2$ is $f^{-1}\left(x\right)=\frac{x-2}{4}$. Graph is shown below.

Step 1. Given information.

Given a one-to-one function $f\left(x\right)=4x+2$.

Step 2. Find the inverse of the function f and verify .

To find the explicit form of the inverse solve $x=4y+2$ for y.

It can be seen that $y=\frac{x-2}{4}$.

Note that if $f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$, then f and g are inverse of one another.

Prove $f\left(f^{-1}\left(x\right)\right)=x$ as follows.

$\begin{array}{rcl}f\left(f^{-1}\left(x\right)\right)& =& f\left(\frac{x-2}{4}\right)\\ & =& 4\left(\frac{x-2}{4}\right)+2\\ & =& x-2+2\\ & =& x\end{array}$

Prove $f^{-1}\left(f\left(x\right)\right)=x$ as follows.

$\begin{array}{rcl}{f}^{-1}\left(f\left(x\right)\right)& =& f^{-1}\left(4x+2\right)\\ & =& \frac{4x+2-2}{4}\\ & =& x\end{array}$

Therefore, $f^{-1}\left(x\right)=\frac{x-2}{4}$ is inverse of $f\left(x\right)=4x+2$.

Step 3. Graph f , f - 1 and y = x on the same coordinate axes.

Graph $f\left(x\right)=4x+2,f^{-1}\left(x\right)=\frac{x-2}{4}$ and $y=x$ as follows.

Note the symmetry of the graphs with respect to the line $y=x$.