Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![Title: Solving for the Inverse Function
Content:
In this example, we are given a function and asked to determine its inverse. The function is defined as:
\[ f(x) = \sqrt[7]{7x - 6}. \]
We need to find the inverse of this function, denoted as \( f^{-1}(x) \).
To solve for \( f^{-1}(x) \), we follow these general steps:
1. **Replace \( f(x) \) with \( y \):**
\[ y = \sqrt[7]{7x - 6}. \]
2. **Solve for \( x \) in terms of \( y \):**
Begin by raising both sides to the power of 7 to eliminate the seventh root:
\[ y^7 = 7x - 6. \]
3. **Isolate \( x \):**
\[ 7x = y^7 + 6. \]
\[ x = \frac{y^7 + 6}{7}. \]
4. **Exchange \( x \) and \( y \) to express \( f^{-1}(x) \):**
\[ f^{-1}(x) = \frac{x^7 + 6}{7}. \]
Therefore, the inverse function is:
\[ f^{-1}(x) = \frac{x^7 + 6}{7}. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4e7be12-e868-44b9-86d4-0a000fb9596b%2Ffd24554d-9779-43a7-87c6-1cdf1d7c1696%2F5n6g69f_processed.png&w=3840&q=75)
Transcribed Image Text:Title: Solving for the Inverse Function
Content:
In this example, we are given a function and asked to determine its inverse. The function is defined as:
\[ f(x) = \sqrt[7]{7x - 6}. \]
We need to find the inverse of this function, denoted as \( f^{-1}(x) \).
To solve for \( f^{-1}(x) \), we follow these general steps:
1. **Replace \( f(x) \) with \( y \):**
\[ y = \sqrt[7]{7x - 6}. \]
2. **Solve for \( x \) in terms of \( y \):**
Begin by raising both sides to the power of 7 to eliminate the seventh root:
\[ y^7 = 7x - 6. \]
3. **Isolate \( x \):**
\[ 7x = y^7 + 6. \]
\[ x = \frac{y^7 + 6}{7}. \]
4. **Exchange \( x \) and \( y \) to express \( f^{-1}(x) \):**
\[ f^{-1}(x) = \frac{x^7 + 6}{7}. \]
Therefore, the inverse function is:
\[ f^{-1}(x) = \frac{x^7 + 6}{7}. \]
Expert Solution
Step 1
Given:-
f(x) = (7x-6)(1/7)
To find:-
Inverse of given function
Step by step
Solved in 3 steps with 2 images
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