Chapter 3.3, Problem 70E

Solution

To explain : The comparative analysis of domain, range, intercepts and asymptotes of the given function and graph them in the manner of Exploration 1.

The domain of the function is the range of its inverse and vice versa. The intercept of the function is $\left(0,1\right)$ and of its inverse is $\left(1,0\right)$ and the asymptotes of the function is $y=0$ and of its inverse is $x=0$ .

Given information :

The functions: $f\left(x\right)=5^x$ and $f^{-1}\left(x\right)={\log }_{5}x$ .

Explanation :

The graph of the given function and its inverse on the same coordinate axis is given below:

  

On the basis of the graph above, the domain, range, intercepts and asymptotes are shown below:

	$f\left(x\right)=5^x$	$f^{-1}\left(x\right)={\log }_{5}x$
Domain	$\left(-\infty ,\infty \right)$	$\left(0,\infty \right)$
Range	$\left(0,\infty \right)$	$\left(-\infty ,\infty \right)$
Intercepts	$\left(0,1\right)$	$\left(1,0\right)$
Asymptotes	$y=0$	$x=0$

Therefore, the domain of the function is the range of its inverse and vice versa. The intercept of the function is $\left(0,1\right)$ and of its inverse is $\left(1,0\right)$ and the asymptotes of the function is $y=0$ and of its inverse is $x=0$ .