Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 9

Chapter 3, Problem 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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

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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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Use the graph of y = f(x) to graph each function g.

g(x) = -f(x) +3

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Use the graph of y = f(x) to graph each function g.

g(x) = f(-x)+3

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