Question

given the function f(x) = (x + 8)^3, complete parts a through c. (a) find an equation for f⁻¹(x). (b) graph f and f⁻¹ in the same rectangular coordinate system. (c) use interval notation to give the domain and the range of f and f⁻¹. (a) find f⁻¹(x). f⁻¹(x) = \square (type an exact answer, using radicals as needed.)

Explanation:

Step1: Swap x and y

Let \( y = (x + 8)^3 \). Swap \( x \) and \( y \) to get \( x = (y + 8)^3 \).

Step2: Solve for y

Take the cube root of both sides: \( \sqrt[3]{x} = y + 8 \). Then subtract 8 from both sides: \( y = \sqrt[3]{x} - 8 \). So \( f^{-1}(x) = \sqrt[3]{x} - 8 \).

Answer:

\( \sqrt[3]{x} - 8 \)