Q 57. - Precalculus Enhanced with Graphing Utilities

Question

The function $f\left(x\right)=\frac{4}{x}$ 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 $f\left(x\right)=\frac{4}{x}$ is $f^{-1}\left(x\right)=\frac{4}{x}$. Graph is shown below.

Step 1. Given information.

Given a one-to-one function $f\left(x\right)=\frac{4}{x}$.

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

To find the explicit form of the inverse solve $x=\frac{4}{y}$ for y.

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

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.

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

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

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

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

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

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

Note that function f and its inverse is one and the same. So, the graph for both coincides.

Broken blue color line represents the function f and unbroken orange color line represents its inverse.