Question

Find the inverse function of \(f\). Use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$ f(x)=\frac{x+2}{x} $$

Short answer

The inverse function of \(f(x) = \frac{x + 2}{x}\) is \(f^{-1}(x) = \frac{2}{x-1}\). The graph of the function and its inverse are reflections of each other over the line \(y = x\).

Step-by-step solution

Rearrange the function to isolate y

To find the inverse function, we must first express the given function in the form \(y = \frac{x + 2}{x}\). This makes \(y\) the subject of the function and makes it easier to solve for its inverse.

Swap x and y

Then, we exchange the roles of \(x\) and \(y\). This gives us the equation \(x = \frac{y + 2}{y}\). According to the definition of the inverse function, if \(y = f(x)\) then \(x = f^{-1}(y)\). Thus, swapping \(x\) and \(y\) permits us to find the inverse function.

Solve for y

In order to find \(f^{-1}(x)\), solve the equation \(x = \frac{y + 2}{y}\) for \(y\). This yield the function \(y = \frac{2}{x-1}\). Hence, \(f^{-1}(x) = \frac{2}{x-1}\).

Graphing and Relationship

Upon graphing \(f(x)\) and \(f^{-1}(x)\) on the same axes, it becomes evident that they are reflected over the line \(y = x\). This symmetry over the line \(y = x\) is a signature characteristic of functions and their inverses.