Question

In Exercises \(13-24,\) find \(f^{-1}\) and verify that $$\left(f \circ f^{-1}\right)(x)=\left(f^{-1} \circ f\right)(x)=x$$ \(f(x)=x^{2 / 3}, \quad x \geq 0\)

Short answer

The inverse function \(f^{-1}(x)\) of the given function \(f(x)=x^{2/3}\) is \(f^{-1}(x) = \sqrt{x^3}\) for \(x\geq0\). The properties of the inverse function have been verified as \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\) for every \(x\geq0\).

Step-by-step solution

Find the Inverse Function

Given \(f(x)=x^{2 / 3}, \quad x \geq 0\), we swap \(x\) and \(f(x)\) to find the inverse function, i.e., \(x=y^{2/3}\), where \(y=f^{-1}(x)\). We solve this for \(y\) by cubing both sides, yielding \(x^3=y^2\), and then take the square root of both sides, to find \(y=f^{-1}(x)=\sqrt{x^3}\), for \(x\geq0\).

Verify the Inverse Function

Substitute \(x=f^{-1}(y)\) into \(f(x)\) which yields the function \(f \circ f^{-1}(x)=f(\sqrt{x^3})=(\sqrt{x^3})^{2/3} = x\) for all \(x\geq0\). Substitute \(x=f(x)\) into \(f^{-1}(x)\) which yields the function \(\left(f^{-1} \circ f\right)(x) = f^{-1}((x^{2/3})^3) = f^{-1}(x^2) = \sqrt{(x^2)^3} = x\) for all \(x\geq0\). This verifies the properties of the inverse function according to the given criteria.