Find the inverse function of h(x) = x^3.

• A. h^{-1}(x) = x^{1/3}
• B. h^{-1}(x) = x^3
• C. h^{-1}(x) = 1/x^3
• D. h^{-1}(x) = 3x^2

Answer: (A)

When a function is inverted, you’re asking for a rule that reverses what the original rule does. For h(x) = x^3, the inverse should undo cubing. Start with y = x^3 and solve for x: x^3 = y implies x = ∛y. Replace y with x to express the inverse as a function of x: h^{-1}(x) = ∛x (which is x^(1/3)).

This works for all real x because the cube root is defined for every real number, and x^3 is strictly increasing on the real line, so it has a unique inverse. So the inverse function is the cube root function.

The other forms don’t fit as the inverse: they don’t undo cubing (the original function itself, the reciprocal 1/x^3, and the derivative 3x^2).