Question

given the function f(x) = (x + 10)^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: Replace \( f(x) \) with \( y \)

We start with the function \( f(x)=(x + 10)^3 \). Replace \( f(x) \) with \( y \), so we have \( y=(x + 10)^3 \).

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

To find the inverse function, we swap the roles of \( x \) and \( y \). This gives us \( x=(y + 10)^3 \).

Step3: Solve for \( y \)

We need to isolate \( y \). Take the cube - root of both sides of the equation \( x=(y + 10)^3 \). The cube - root of \( x \) is \( \sqrt[3]{x} \), and the cube - root of \( (y + 10)^3 \) is \( y + 10 \). So we get \( \sqrt[3]{x}=y + 10 \). Then, subtract 10 from both sides to solve for \( y \): \( y=\sqrt[3]{x}-10 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

We replace \( y \) with \( f^{-1}(x) \), so \( f^{-1}(x)=\sqrt[3]{x}-10 \).

Answer:

\( f^{-1}(x)=\sqrt[3]{x}-10 \)