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Find f(x) for the function. X x + 1 f(x) = f-1(x) =

Find f(x) for the function. X x + 1 f(x) = f-1(x) =

Algebra and Trigonometry (6th Edition)
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.
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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