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For the function f, use composition of functions to show that f^-1 (the inverse) is as given

For the function f, use composition of functions to show that f^-1 (the inverse) is as given

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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For the function f, use composition of functions to show that f^-1 (the inverse) is as given
### Understanding Inverse Functions #### Problem Statement **78.** Given the function \( f(x) = \frac{x + 6}{3x - 4} \), its inverse \( f^{-1}(x) \) is provided as \( \frac{4x + 6}{3x - 1} \). #### Explanation The task involves understanding how the function \( f(x) \) maps values, and how its inverse function \( f^{-1}(x) \) operates to reverse that mapping. - **Original Function:** \( f(x) = \frac{x + 6}{3x - 4} \) This function takes an input \( x \), processes it through a rational expression where the numerator is \( x + 6 \) and the denominator is \( 3x - 4 \). - **Inverse Function:** \( f^{-1}(x) = \frac{4x + 6}{3x - 1} \) The inverse function undoes the operation of \( f(x) \). For any output of \( f(x) \), when plugged into \( f^{-1}(x) \), it should return the original input \( x \). #### Importance of Inverse Functions Inverse functions are critical in mathematics as they allow us to 'reverse' a function's effect. In other words, they specify how to get back to the original input value given the output of a function. Understanding and verifying the correctness of inverse functions requires ensuring that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) for all eligible values of \( x \). This exercise is particularly useful in fields like calculus, algebra, and trigonometry, providing foundational skills for dealing with transformations and mappings in mathematical systems.
Transcribed Image Text:### Understanding Inverse Functions #### Problem Statement **78.** Given the function \( f(x) = \frac{x + 6}{3x - 4} \), its inverse \( f^{-1}(x) \) is provided as \( \frac{4x + 6}{3x - 1} \). #### Explanation The task involves understanding how the function \( f(x) \) maps values, and how its inverse function \( f^{-1}(x) \) operates to reverse that mapping. - **Original Function:** \( f(x) = \frac{x + 6}{3x - 4} \) This function takes an input \( x \), processes it through a rational expression where the numerator is \( x + 6 \) and the denominator is \( 3x - 4 \). - **Inverse Function:** \( f^{-1}(x) = \frac{4x + 6}{3x - 1} \) The inverse function undoes the operation of \( f(x) \). For any output of \( f(x) \), when plugged into \( f^{-1}(x) \), it should return the original input \( x \). #### Importance of Inverse Functions Inverse functions are critical in mathematics as they allow us to 'reverse' a function's effect. In other words, they specify how to get back to the original input value given the output of a function. Understanding and verifying the correctness of inverse functions requires ensuring that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) for all eligible values of \( x \). This exercise is particularly useful in fields like calculus, algebra, and trigonometry, providing foundational skills for dealing with transformations and mappings in mathematical systems.
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