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
The function f is one-to-one. Find its inverse.
![**Educational Content on Inverse Functions**
**Problem Statement**
The function \( f \) is one-to-one. Find its inverse.
\[ f(x) = \frac{4}{x - 5}, \quad x \neq 5 \]
**Options for the Inverse Function**
- **A.** \( f^{-1}(x) = \frac{-5 + 4x^2}{x}, \quad x \neq 0 \)
- **B.** \( f^{-1}(x) = \frac{-5 + 4x}{x}, \quad x \neq 0 \)
- **C.** \( f^{-1}(x) = \frac{x}{-5 + 4x}, \quad x \neq \frac{5}{4} \)
**Explanation**
In this problem, the task is to determine the inverse function \( f^{-1}(x) \) for the given one-to-one function \( f(x) = \frac{4}{x - 5} \) where \( x \neq 5 \). The function is a rational expression, and the options given represent different formulations of potential inverse functions.
Each option includes an expression to test whether it correctly represents the inverse of the given function, and specifies domains to exclude values that would cause division by zero.
**Understanding the Graph or Diagram**
This task does not include any graphs or diagrams, but focuses on ensuring that within the constraints given for the function \( f(x) \) (i.e., \( x \neq 5 \)) and for the inverse candidates \( f^{-1}(x) \), the calculations for the inverse function should hold true across valid domains. An inverse function effectively "undoes" the operation of the original function across its domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d66f0ae-8c6e-4ff5-9763-c8b27ba74e97%2Fe4bf6795-8861-483b-ab05-d2cee511d682%2Fz1q1hqe_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content on Inverse Functions**
**Problem Statement**
The function \( f \) is one-to-one. Find its inverse.
\[ f(x) = \frac{4}{x - 5}, \quad x \neq 5 \]
**Options for the Inverse Function**
- **A.** \( f^{-1}(x) = \frac{-5 + 4x^2}{x}, \quad x \neq 0 \)
- **B.** \( f^{-1}(x) = \frac{-5 + 4x}{x}, \quad x \neq 0 \)
- **C.** \( f^{-1}(x) = \frac{x}{-5 + 4x}, \quad x \neq \frac{5}{4} \)
**Explanation**
In this problem, the task is to determine the inverse function \( f^{-1}(x) \) for the given one-to-one function \( f(x) = \frac{4}{x - 5} \) where \( x \neq 5 \). The function is a rational expression, and the options given represent different formulations of potential inverse functions.
Each option includes an expression to test whether it correctly represents the inverse of the given function, and specifies domains to exclude values that would cause division by zero.
**Understanding the Graph or Diagram**
This task does not include any graphs or diagrams, but focuses on ensuring that within the constraints given for the function \( f(x) \) (i.e., \( x \neq 5 \)) and for the inverse candidates \( f^{-1}(x) \), the calculations for the inverse function should hold true across valid domains. An inverse function effectively "undoes" the operation of the original function across its domain.
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