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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![### Finding the Inverse of a Given Function
This exercise involves determining the inverse of the given function. The function provided is:
\[ f(x) = \frac{4x - 3}{x + 1} \]
#### Steps to Find the Inverse:
1. **Replace \( f(x) \) with \( y \):**
\[ y = \frac{4x - 3}{x + 1} \]
2. **Swap \( y \) and \( x \) to find the inverse:**
\[ x = \frac{4y - 3}{y + 1} \]
3. **Solve for \( y \):**
Begin by eliminating the fraction by multiplying both sides of the equation by \( (y + 1) \):
\[ x(y + 1) = 4y - 3 \]
Distribute \( x \):
\[ xy + x = 4y - 3 \]
Rearrange the equation to group the \( y \)-terms on one side and the non-\( y \)-terms on the other side:
\[ xy - 4y = - x - 3 \]
Factor out \( y \) from the left side:
\[ y(x - 4) = - x - 3 \]
Solve for \( y \):
\[ y = \frac{- x - 3}{x - 4} \]
4. **Replace \( y \) with \( f^{-1}(x) \):**
\[ f^{-1}(x) = \frac{- x - 3}{x - 4} \]
Therefore, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{- x - 3}{x - 4} \]
#### Summary:
To find the inverse of the function \( f(x) = \frac{4x - 3}{x + 1} \), follow the steps:
1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \).
3. Solve for \( y \).
4. Replace \( y \) with \( f^{-1}(x) \).
The inverse function is \( f^{-1}(x) = \frac{-x - 3}{x -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e5a8746-5172-4f40-b353-4538d9b50312%2F59e447e7-4e71-491b-9c81-b84844ecc961%2Fti00syo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Inverse of a Given Function
This exercise involves determining the inverse of the given function. The function provided is:
\[ f(x) = \frac{4x - 3}{x + 1} \]
#### Steps to Find the Inverse:
1. **Replace \( f(x) \) with \( y \):**
\[ y = \frac{4x - 3}{x + 1} \]
2. **Swap \( y \) and \( x \) to find the inverse:**
\[ x = \frac{4y - 3}{y + 1} \]
3. **Solve for \( y \):**
Begin by eliminating the fraction by multiplying both sides of the equation by \( (y + 1) \):
\[ x(y + 1) = 4y - 3 \]
Distribute \( x \):
\[ xy + x = 4y - 3 \]
Rearrange the equation to group the \( y \)-terms on one side and the non-\( y \)-terms on the other side:
\[ xy - 4y = - x - 3 \]
Factor out \( y \) from the left side:
\[ y(x - 4) = - x - 3 \]
Solve for \( y \):
\[ y = \frac{- x - 3}{x - 4} \]
4. **Replace \( y \) with \( f^{-1}(x) \):**
\[ f^{-1}(x) = \frac{- x - 3}{x - 4} \]
Therefore, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{- x - 3}{x - 4} \]
#### Summary:
To find the inverse of the function \( f(x) = \frac{4x - 3}{x + 1} \), follow the steps:
1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \).
3. Solve for \( y \).
4. Replace \( y \) with \( f^{-1}(x) \).
The inverse function is \( f^{-1}(x) = \frac{-x - 3}{x -
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