Question

In Exercises 101 and \(102,\) use a graphing utility to confirm that \(f\) and \(g\) are inverse functions. (Remember to restrict the domain of \(f\) properly. $$ \begin{array}{l} f(x)=\tan x \\ g(x)=\arctan x \end{array} $$

Short answer

If the plots of the functions \(f(x) = \tan x\) and \(g(x) = \arctan x\) are reflections of each other across the line \(y = x\), then the two functions are indeed inverse functions.

Step-by-step solution

Plotting the Tangent Function

The first step is to plot the tangent function \(f(x) = \tan x\) using a graphing utility. The x-values should be restricted to \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) so it is strictly increasing and to match the range of the \(\arctan x\) function.

Plotting the Arctangent Function

Next, plot the arctangent function \(g(x) = \arctan x\) on the same graph. No restriction is required for this function, as the domain is all real numbers.

Reflection Analysis

Look at the two plots. If the functions \(f(x)\) and \(g(x)\) are inverse functions, they should be reflections of each other across the line \(y = x\).