91Ó°ÊÓ

Plug \(g(x)\) into \(f(x)\) and simplify. Then, plug \(f(x)\) into \(g(x)\) and simplify. This should yield \(x\) in both cases if \(f\) and \(g\) are indeed inverses of each other. \nSo, first compute \(f(g(x))\): Start with \(f(x)\) as defined, but wherever \(x\) appears, replace it with \(g(x)\). So it becomes \(f(g(x)) = \left(\sqrt[3]{8x}\right)^3/8\). Simplifying this expression, we get \(f(g(x)) = x\). Next, compute \(g(f(x))\): Start with \(g(x)\) as defined, but wherever \(x\) appears, replace it with \(f(x)\). So it becomes \(g(f(x)) = \sqrt[3]{8*(x^3/8)}\). Simplifying this expression, we get \(g(f(x)) = x\). As both \(f(g(x))\) and \(g(f(x))\) yield the original input \(x\), this confirms algebraically that \(f\) and \(g\) are inverses of each other.

Plot the functions \(f(x)\), \(g(x)\), and \(y = x\) on the same graph. Since \(f(x) = x^3/8\) is a cubic function and \(g(x) = \sqrt[3]{8x}\) is a cubic root function, their graphs would be upward and rising. The line \(y = x\) is a straight line with a slope of 1. If \(f\) and \(g\) are inverses, then their graphs should be mirror images with respect to the line \(y = x\). Upon checking, one would observe that the graphs of \(f(x)\) and \(g(x)\) are indeed symmetric about this line, confirming that \(f\) and \(g\) are inversely related to each other graphically.