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 Function
To find the inverse function \( f^{-1}(x) \) for the given function:
\[ f(x) = \frac{x}{x + 1} \]
We need to solve for \( x \) in terms of \( y \) and then interchange \( x \) and \( y \).
### Steps to Find the Inverse:
1. **Start with the given function and replace \( f(x) \) with \( y \):**
\[ y = \frac{x}{x + 1} \]
2. **Solve for \( x \) in terms of \( y \):**
Multiply both sides by \( x + 1 \):
\[ y(x + 1) = x \]
Distribute \( y \):
\[ yx + y = x \]
Rearrange to isolate terms involving \( x \):
\[ yx - x = -y \]
Factor out \( x \):
\[ x(y - 1) = -y \]
Solve for \( x \):
\[ x = \frac{-y}{y - 1} \]
3. **Interchange \( x \) and \( y \) to find the inverse function \( f^{-1}(x) \):**
\[ f^{-1}(x) = \frac{-x}{x - 1} \]
Therefore, the inverse function is:
\[ f^{-1}(x) = \frac{-x}{x - 1} \]
This method involves algebraic manipulation and understanding of function inverses. By solving \( y = \frac{x}{x+1} \) for \( x \), we derive the inverse function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4895c6df-eb5a-4830-9b1e-b00e9a38873d%2Fc3d2c9fa-aa1b-4778-926b-fcf953be6ece%2Fpc4oeah_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Inverse Function
To find the inverse function \( f^{-1}(x) \) for the given function:
\[ f(x) = \frac{x}{x + 1} \]
We need to solve for \( x \) in terms of \( y \) and then interchange \( x \) and \( y \).
### Steps to Find the Inverse:
1. **Start with the given function and replace \( f(x) \) with \( y \):**
\[ y = \frac{x}{x + 1} \]
2. **Solve for \( x \) in terms of \( y \):**
Multiply both sides by \( x + 1 \):
\[ y(x + 1) = x \]
Distribute \( y \):
\[ yx + y = x \]
Rearrange to isolate terms involving \( x \):
\[ yx - x = -y \]
Factor out \( x \):
\[ x(y - 1) = -y \]
Solve for \( x \):
\[ x = \frac{-y}{y - 1} \]
3. **Interchange \( x \) and \( y \) to find the inverse function \( f^{-1}(x) \):**
\[ f^{-1}(x) = \frac{-x}{x - 1} \]
Therefore, the inverse function is:
\[ f^{-1}(x) = \frac{-x}{x - 1} \]
This method involves algebraic manipulation and understanding of function inverses. By solving \( y = \frac{x}{x+1} \) for \( x \), we derive the inverse function.
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