Question

Find the inverse function of \(f\). Graph (by hand) \(f\) and \(f^{-1}\). Describe the relationship between the graphs. $$ f(x)=\sqrt{x} $$

Short answer

The inverse function of \(f(x) = \sqrt{x}\) is \(f^{-1}(x) = x^2\). Both the functions' graphs are reflections of each other across the line \(y = x\).

Step-by-step solution

Find the inverse function

Start off by replacing \(f(x)\) with \(y\). Our equation now becomes \(y = \sqrt{x}\). To find the inverse, the roles of x and y need to be switched, turning into \(x = \sqrt{y}\). Solving for \(y\), yields \(y = x^2\). Therefore, the inverse function of \(f\), denoted as \(f^{-1}(x)\), is \(f^{-1}(x) = x^2\).

Graph the functions

Now, we need to sketch the graph. The graph of \(f(x) = \sqrt{x}\) is a line that starts at the origin (0, 0) and extends upwards and to the right; the part of the parabola lying in the first quadrant. The graph of \(f^{-1}(x) = x^2\) is a parabola opening upwards, including both positive and negative x-values.

Describe the relationship between the graphs

We can observe that the graph of \(f^{-1}\) is a reflection of the graph of \(f\) across the line \(y = x\). This is always true for a function and its inverse. They are symmetrical to the line \(y = x\)