College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 82E
Related questions
Question
![**Title: Finding the Inverse Function**
**Problem Statement:**
Find the inverse function of \( f(x) = 12 + \sqrt[3]{x} \).
\[ f^{-1}(x) = \underline{\hspace{2cm}} \]
**Explanation:**
To find the inverse function of \( f(x) \), we follow these steps:
1. **Replace \( f(x) \) with \( y \):**
\[ y = 12 + \sqrt[3]{x} \]
2. **Swap \( x \) and \( y \) to find the inverse:**
\[ x = 12 + \sqrt[3]{y} \]
3. **Solve for \( y \) in terms of \( x \):**
Subtract 12 from both sides:
\[ x - 12 = \sqrt[3]{y} \]
Cube both sides to solve for \( y \):
\[ (x - 12)^3 = y \]
4. **Write the inverse function:**
\[ f^{-1}(x) = (x - 12)^3 \]
Therefore, the inverse function is:
\[ f^{-1}(x) = (x - 12)^3 \]
This procedure helps you derive the inverse of the given function. In this case, the function's inverse is a cubic function, transforming the variable \( x \) after subtracting 12 and then cubing the result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa21e1c50-d204-4b1c-af4d-5bfcd54fd78e%2Fef0b5858-d85c-4dbf-9a34-3058614d9061%2Fixg7qu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Inverse Function**
**Problem Statement:**
Find the inverse function of \( f(x) = 12 + \sqrt[3]{x} \).
\[ f^{-1}(x) = \underline{\hspace{2cm}} \]
**Explanation:**
To find the inverse function of \( f(x) \), we follow these steps:
1. **Replace \( f(x) \) with \( y \):**
\[ y = 12 + \sqrt[3]{x} \]
2. **Swap \( x \) and \( y \) to find the inverse:**
\[ x = 12 + \sqrt[3]{y} \]
3. **Solve for \( y \) in terms of \( x \):**
Subtract 12 from both sides:
\[ x - 12 = \sqrt[3]{y} \]
Cube both sides to solve for \( y \):
\[ (x - 12)^3 = y \]
4. **Write the inverse function:**
\[ f^{-1}(x) = (x - 12)^3 \]
Therefore, the inverse function is:
\[ f^{-1}(x) = (x - 12)^3 \]
This procedure helps you derive the inverse of the given function. In this case, the function's inverse is a cubic function, transforming the variable \( x \) after subtracting 12 and then cubing the result.
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