91Ó°ÊÓ

Write the given function by replacing f(x) with y: $$ y = \cot^{-1} \left(\frac{x}{3}\right) $$

Now, swap x and y in the equation: $$ x = \cot^{-1} \left(\frac{y}{3}\right) $$

To solve for y, first, apply cotangent on both sides of the equation: $$ \cot(x) = \frac{y}{3} $$ Now, multiply both sides of the equation by 3 to get y: $$ y = 3\cot(x) $$ Therefore, the inverse function is: $$ f^{-1}(x) = 3\cot(x) $$

In order to sketch the graph of \(f(x) = \cot^{-1}\left(\frac{x}{3}\right)\) and its inverse \(f^{-1}(x) = 3\cot(x)\), follow these guidelines: 1. Sketch the graph of the cotangent function (\(\cot(x)\)). Remember that the cotangent function has vertical asymptotes at \(x = k\pi\) for all integer values of k and has a period of \(\pi\). 2. For the given function \(f(x)\), scale the cotangent graph down by a factor of 3 vertically. 3. For the inverse function \(f^{-1}(x)\), scale the cotangent graph up by a factor of 3 horizontally. 4. Finally, make sure that the given function and its inverse are reflections of each other across the line (\(y = x\)). By following these guidelines, you'll be able to sketch the graph of the function and its inverse on the same set of axes.