Question

Find the inverse function of \(f\). Graph (by hand) \(f\) and \(f^{-1}\). Describe the relationship between the graphs. $$ f(x)=x^{2}, \quad x \geq 0 $$

Short answer

The inverse function of \(f(x)=x^{2}\) is \(f^{-1}(x) = \sqrt{x}\).

Step-by-step solution

Find the inverse function

To find the inverse of the function \(f(x)=x^{2}\), replace \(f(x)\) with \(y\), so we get \(y = x^{2}\). Then, switch \(x\) and \(y\), this gives \(x = y^{2}\). Solving for \(y\) gives \(y = \sqrt{x}\) which is the inverse function \(f^{-1}(x)=\sqrt{x}\).

Graphing the function \(f(x)=x^{2}\) and its inverse \(f^{-1}(x)=\sqrt{x}\)

A quadratic function \(f(x)=x^2\) opens upwards with a vertex at the origin (0,0), whereas its inverse, the square root function \(f^{-1}(x)=\sqrt{x}\), looks like the right half of a horizontal parabola. They are reflections of each other in the line \(y=x\).

Relationship between the graphs

Since every function and its inverse are reflections of each other over the line \(y=x\), then each x-coordinate on function \(f\) corresponds to the y-coordinate on \(f^{-1}\), and vice versa. Hence, every point (a, b) on the graph of \(f\) corresponds to the point (b, a) on the graph of \(f^{-1}\).