https://file.aiquickdraw.com/mathbot/file/1725854905899mkhmtyju.jpg

The problem in the image is:

"The function \( f(x) = x^3 - 9 \) is one-to-one.
   
a. Find an equation for \( f^{-1}(x) \), the inverse function.

b. Verify that your equation is correct by showing that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)."

### Solution:

#### Part (a): Finding the inverse function \( f^{-1}(x) \)
Given the function \( f(x) = x^3 - 9 \), we want to find its inverse.

1. Start by setting \( y = f(x) \):
   \[
   y = x^3 - 9
   \]
   
2. Now, solve for \( x \) in terms of \( y \):
   \[
   y + 9 = x^3
   \]
   \[
   x = \sqrt[3]{y + 9}
   \]

3. So, the inverse function is:
   \[
   f^{-1}(x) = \sqrt[3]{x + 9}
   \]

#### Part (b): Verifying the inverse
We need to verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

1. **First, verify \( f(f^{-1}(x)) = x \):**
   \[
   f(f^{-1}(x)) = f\left( \sqrt[3]{x + 9} \right)
   \]
   Substitute into the function \( f(x) = x^3 - 9 \):
   \[
   = \left( \sqrt[3]{x + 9} \right)^3 - 9
   \]
   \[
   = (x + 9) - 9
   \]
   \[
   = x
   \]
   This verifies \( f(f^{-1}(x)) = x \).

2. **Next, verify \( f^{-1}(f(x)) = x \):**
   \[
   f^{-1}(f(x)) = f^{-1}(x^3 - 9)
   \]
   Substitute into the inverse function \( f^{-1}(x) = \sqrt[3]{x + 9} \):
   \[
   = \sqrt[3]{(x^3 - 9) + 9}
   \]
   \[
   = \sqrt[3]{x^3}
   \]
   \[
   = x
   \]
   This verifies \( f^{-1}(f(x)) = x \).

Thus, the inverse function is \( f^{-1}(x) = \sqrt[3]{x + 9} \), and the verification holds.

Would you like further details on any step or have questions?

Here are 5 related questions to explore further:
1. How can we verify if a function is one-to-one without finding the inverse?
2. Can the inverse of a function always be found? Why or why not?
3. What is the geometric interpretation of the inverse function?
4. How does the derivative of a function relate to its inverse?
5. What are the conditions for a cubic function to have a unique inverse?

**Tip:** To verify if a function has an inverse, check if the function is **one-to-one** using the horizontal line test!

The function f(x) = x^3 - 9 is one-to-one.

a. Find an equation for f^{-1}(x), the inverse function.

b. Verify that your equation is correct by showing that f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

Learn how to find the inverse of the cubic function f(x) = x^3 - 9 and verify the correctness of the inverse. This algebra problem includes step-by-step solutions, covering inverse functions and verification techniques for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation