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