Are the following functions inverse \( f(x)=\sqrt[3]{x-5} \) \( g(x)=x^{3}+5 \)

To determine if the functions \( f(x) = \sqrt[3]{x-5} \) and \( g(x) = x^3 + 5 \) are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \). Starting with \( f(g(x)) \): \[ f(g(x)) = f(x^3 + 5) = \sqrt[3]{(x^3 + 5) - 5} = \sqrt[3]{x^3} = x \] Now, checking \( g(f(x)) \): \[ g(f(x)) = g(\sqrt[3]{x-5}) = (\sqrt[3]{x-5})^3 + 5 = x - 5 + 5 = x \] Since both compositions yield \( x \), the functions \( f \) and \( g \) are indeed inverses of each other! The relationship between a function and its inverse is a beautiful part of mathematics; it shows that each function undoes the other, reflecting the symmetry between operations. If you’re interested in how these functions fit into the broader landscape of functions, consider exploring concepts of transformations, where shifts, reflections, and stretches apply to parent functions. It expands your understanding of function behavior and their relationships!