Domanda del libro di testo
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(1, 4), (1, 5), (1, 6)}
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 3, Problema 9
Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = = -x and g(x) = -x
Guida verificata passo dopo passo1
Identify the given functions: \(f(x) = -x\) and \(g(x) = -x\).
Find the composition \(f(g(x))\) by substituting \(g(x)\) into \(f\): write \(f(g(x)) = f(-x)\).
Evaluate \(f(-x)\) by replacing the input of \(f\) with \(-x\): since \(f(t) = -t\), then \(f(-x) = -(-x)\).
Simplify \(f(g(x))\): \(-(-x) = x\).
Similarly, find \(g(f(x))\) by substituting \(f(x)\) into \(g\): write \(g(f(x)) = g(-x)\), then evaluate \(g(-x) = -(-x) = x\). Since both compositions equal \(x\), conclude that \(f\) and \(g\) are inverses of each other.
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Function Composition
Function composition involves applying one function to the result of another, denoted as f(g(x)). It means substituting g(x) into f(x), which helps analyze how two functions interact and combine their effects.
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Function Composition
Inverse Functions
Inverse functions reverse each other's operations, so f(g(x)) = x and g(f(x)) = x for all x in the domain. Identifying inverses requires checking if composing the functions in both orders returns the original input.
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Properties of Linear Functions
Linear functions have the form f(x) = mx + b. Understanding their behavior, especially when m = -1 and b = 0 as in f(x) = -x, is essential for evaluating compositions and determining if two linear functions are inverses.
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