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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 79a
Chapter 3, Problem 79a

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.
ƒ(x)=√(x−1),g(x)=3xƒ(x)=√(x-1), g(x)=3x

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1
Identify the given functions: \(f(x) = \sqrt{x - 1}\) and \(g(x) = 3x\).
Find the composition \((f \circ g)(x)\), which means \(f(g(x))\). Substitute \(g(x)\) into \(f\): \((f \circ g)(x) = f(3x) = \sqrt{3x - 1}\).
Determine the domain of \((f \circ g)(x)\) by considering the domain restrictions of \(f\) after substitution. Since \(f(x) = \sqrt{x - 1}\) requires \(x - 1 \geq 0\), set the inside of the square root in the composition \(3x - 1 \geq 0\).
Solve the inequality \(3x - 1 \geq 0\) to find the domain of \((f \circ g)(x)\). This gives \(x \geq \frac{1}{3}\).
Conclude that the domain of \((f \circ g)(x)\) is all real numbers \(x\) such that \(x \geq \frac{1}{3}\).

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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) = f(g(x)). It means you first evaluate g at x, then use that output as the input for f. Understanding this process is essential to correctly find (f∘g)(x).
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Function Composition

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When composing functions, the domain of (f∘g) includes all x-values in the domain of g such that g(x) lies in the domain of f. Identifying these restrictions ensures the composition is valid.
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Domain Restrictions of Composed Functions

Square Root Function Domain Restrictions

For functions involving square roots, the expression inside the root must be non-negative to produce real outputs. For f(x) = √(x - 1), the domain requires x - 1 ≥ 0, or x ≥ 1. This restriction affects the domain of the composition when substituting g(x) into f.
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Related Practice
Textbook Question

An equation that defines y as a function of x is given. Find ƒ(3). x-4y=8

2
views
Textbook Question

Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x+2,g(x)=x4+x2−4ƒ(x)=x+2, g(x)=x^4+x^2-4

Textbook Question

An equation that defines y as a function of x is given. Find ƒ(3). y+2x2=3-x

1
views
Textbook Question

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=√(x−1),g(x)=3xƒ(x)=√(x-1), g(x)=3x

Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x+2,g(x)=x4+x2−4ƒ(x)=x+2, g(x)=x^4+x^2-4

Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√(x-2), g(x)=2x