Q 52. - Precalculus Enhanced with Graphing Utilities

Question

The function $f\left(x\right)=1-3x$ 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 $f\left(x\right)=1-3x$ is $f^{-1}\left(x\right)=\frac{1-x}{3}$. Graph is shown below.

Step 1. Given information.

Given a one-to-one function $f\left(x\right)=1-3x$.

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

To find the explicit form of the inverse solve $x=1-3y$ for y.

It can be seen that $y=\frac{1-x}{3}$.

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.

$\begin{array}{rcl}f\left(f^{-1}\left(x\right)\right)& =& f\left(\frac{1-x}{3}\right)\\ & =& 1-3\left(\frac{1-x}{3}\right)\\ & =& 1-1+x\\ & =& x\end{array}$

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

$\begin{array}{rcl}{f}^{-1}\left(f\left(x\right)\right)& =& f^{-1}\left(1-3x\right)\\ & =& \frac{1-\left(1-3x\right)}{3}\\ & =& \frac{3x}{3}\\ & =& x\end{array}$

Therefore, $f^{-1}\left(x\right)=\frac{1-x}{3}$ is inverse of $f\left(x\right)=1-3x$.

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

Graph $f\left(x\right)=1-3x,f^{-1}\left(x\right)=\frac{1-x}{3}$ and $y=x$ as follows.

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