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) = \boxed{} (type an exact answer, using radicals as needed.)

Explanation:

Step1: Let \( y = f(x) \)

We have \( y=(x + 8)^{3} \).

Step2: Swap \( x \) and \( y \)

This gives \( x=(y + 8)^{3} \).

Step3: Solve for \( y \)

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

Answer:

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