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6. Determine the algebraic representation of f-(x) from the tabular representation of ƒ(x) and justify that fx) is odd, even, or neither. -2 -1 2 f(x) -5 -2 1 4 7

6. Determine the algebraic representation of f-(x) from the tabular representation of ƒ(x) and justify that fx) is odd, even, or neither. -2 -1 2 f(x) -5 -2 1 4 7

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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**Problem 6: Inverse Function Analysis** Determine the algebraic representation of \( f^{-1}(x) \) from the tabular representation of \( f(x) \) and justify whether \( f^{-1}(x) \) is odd, even, or neither. **Table Representation:** | \( x \) | \( -2 \) | \( -1 \) | \( 0 \) | \( 1 \) | \( 2 \) | |:------:|:------:|:------:|:----:|:----:|:----:| | \( f(x) \) | \( -5 \) | \( -2 \) | \( 1 \) | \( 4 \) | \( 7 \) | **Analysis:** To find \( f^{-1}(x) \), switch the roles of \( x \) and \( f(x) \). The inverse relations are: - \( f(-2) = -5 \) implies \( f^{-1}(-5) = -2 \) - \( f(-1) = -2 \) implies \( f^{-1}(-2) = -1 \) - \( f(0) = 1 \) implies \( f^{-1}(1) = 0 \) - \( f(1) = 4 \) implies \( f^{-1}(4) = 1 \) - \( f(2) = 7 \) implies \( f^{-1}(7) = 2 \) **Determine Odd, Even or Neither:** An even function satisfies \( f(-x) = f(x) \) for all \( x \) in the domain, and an odd function satisfies \( f(-x) = -f(x) \). - Check for evenness: For none of the inputs \( x = a \), does \( f^{-1}(-a) = f^{-1}(a) \). - Check for oddness: For none of the inputs \( x = a \), does \( f^{-1}(-a) = -f^{-1}(a) \). Since \( f^{-1}(x) \) does not satisfy the conditions for being either even or odd, it is classified as **neither**. This exercise helps illustrate the graphical and algebraic symmetry properties of functions and their inverses.
Transcribed Image Text:**Problem 6: Inverse Function Analysis** Determine the algebraic representation of \( f^{-1}(x) \) from the tabular representation of \( f(x) \) and justify whether \( f^{-1}(x) \) is odd, even, or neither. **Table Representation:** | \( x \) | \( -2 \) | \( -1 \) | \( 0 \) | \( 1 \) | \( 2 \) | |:------:|:------:|:------:|:----:|:----:|:----:| | \( f(x) \) | \( -5 \) | \( -2 \) | \( 1 \) | \( 4 \) | \( 7 \) | **Analysis:** To find \( f^{-1}(x) \), switch the roles of \( x \) and \( f(x) \). The inverse relations are: - \( f(-2) = -5 \) implies \( f^{-1}(-5) = -2 \) - \( f(-1) = -2 \) implies \( f^{-1}(-2) = -1 \) - \( f(0) = 1 \) implies \( f^{-1}(1) = 0 \) - \( f(1) = 4 \) implies \( f^{-1}(4) = 1 \) - \( f(2) = 7 \) implies \( f^{-1}(7) = 2 \) **Determine Odd, Even or Neither:** An even function satisfies \( f(-x) = f(x) \) for all \( x \) in the domain, and an odd function satisfies \( f(-x) = -f(x) \). - Check for evenness: For none of the inputs \( x = a \), does \( f^{-1}(-a) = f^{-1}(a) \). - Check for oddness: For none of the inputs \( x = a \), does \( f^{-1}(-a) = -f^{-1}(a) \). Since \( f^{-1}(x) \) does not satisfy the conditions for being either even or odd, it is classified as **neither**. This exercise helps illustrate the graphical and algebraic symmetry properties of functions and their inverses.
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