Questions: 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). (c) State the domain and range of f and f^(-1) using interval notation. The domain of f(x) is , and the range of f(x) is .

Solution Steps

Given the function \( f(x) = (x + 8)^3 \), we need to find its inverse \( f^{-1}(x) \).

1. Start by setting \( y = f(x) \): \[ y = (x + 8)^3 \]

2. Solve for \( x \) in terms of \( y \): \[ y = (x + 8)^3 \implies \sqrt[3]{y} = x + 8 \implies x = \sqrt[3]{y} - 8 \]

3. Swap \( x \) and \( y \) to get the inverse function: \[ f^{-1}(x) = \sqrt[3]{x} - 8 \]

To graph \( f(x) = (x + 8)^3 \) and \( f^{-1}(x) = \sqrt[3]{x} - 8 \):

1. Plot the function \( f(x) = (x + 8)^3 \):

   • This is a cubic function shifted 8 units to the left.

2. Plot the inverse function \( f^{-1}(x) = \sqrt[3]{x} - 8 \):

   • This is a cube root function shifted 8 units down.

3. Ensure the graphs are reflections of each other across the line \( y = x \).

1. Determine the domain and range of \( f(x) = (x + 8)^3 \):

   • The domain of \( f(x) \) is all real numbers \( (-\infty, \infty) \).
   • The range of \( f(x) \) is all real numbers \( (-\infty, \infty) \).

2. Determine the domain and range of \( f^{-1}(x) = \sqrt[3]{x} - 8 \):

   • The domain of \( f^{-1}(x) \) is all real numbers \( (-\infty, \infty) \).
   • The range of \( f^{-1}(x) \) is all real numbers \( (-\infty, \infty) \).

Final Answer