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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 77a
Chapter 3, Problem 77a

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.
ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1

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1
Identify the given functions: \(f(x) = x^3\) and \(g(x) = x^2 + 3x - 1\).
Recall that the composition of functions \((f \circ g)(x)\) means \(f(g(x))\), which is applying \(g\) first, then \(f\) to the result.
Substitute \(g(x)\) into \(f\): write \((f \circ g)(x) = f(g(x)) = (g(x))^3\).
Express \((f \circ g)(x)\) explicitly by cubing the expression for \(g(x)\): \((x^2 + 3x - 1)^3\).
Determine the domain of \((f \circ g)(x)\) by considering the domain of \(g(x)\) and the domain of \(f\) applied to \(g(x)\). Since both \(f\) and \(g\) are polynomials, their domains are all real numbers, so the domain of \((f \circ g)(x)\) is all real numbers.

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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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Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents with coefficients. Here, f(x) = x^3 and g(x) = x^2 + 3x - 1 are polynomials, which are continuous and defined for all real numbers. Recognizing their form helps in simplifying and composing the functions.
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Domain of Composite Functions

The domain of (f∘g)(x) consists of all x-values in the domain of g for which g(x) is in the domain of f. Since polynomials are defined for all real numbers, the domain is typically all real numbers unless restricted by the composition. Identifying the domain ensures the composite function is valid.
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Related Practice
Textbook Question

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

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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

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Textbook Question

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

ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1

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Textbook Question

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

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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, g(x)=x-1

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