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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 77b
Chapter 3, Problem 77b

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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1
Recall that the composition of functions (g \(\circ\) f)(x) means g(f(x)), which is the function g applied to the output of f(x).
Given f(x) = x^3, substitute f(x) into g(x) to find (g \(\circ\) f)(x). This means replace every x in g(x) with f(x) = x^3.
Write the expression for (g \(\circ\) f)(x) as g(f(x)) = (f(x))^2 + 3(f(x)) - 1, which becomes (x^3)^2 + 3(x^3) - 1.
Simplify the expression by applying the exponent and multiplication: (x^3)^2 = x^{6}, so the expression becomes x^{6} + 3x^{3} - 1.
Determine the domain of (g \(\circ\) f)(x) by considering the domain of f(x) and g(x). Since both f(x) = x^3 and g(x) = x^2 + 3x - 1 are polynomials, their domains are all real numbers, so the domain of (g \(\circ\) f)(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 (g∘f)(x) = g(f(x)). It means you first evaluate f(x), then use that output as the input for g. Understanding this process is essential to correctly form the composite function.
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Evaluating Polynomial Functions

Evaluating polynomial functions requires substituting the input value into the polynomial expression and simplifying. For example, f(x) = x^3 means cubing the input x. This skill is necessary to find g(f(x)) by substituting f(x) into g.
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Introduction to Polynomial Functions

Domain of Composite Functions

The domain of a composite function (g∘f)(x) consists of all x-values in the domain of f for which f(x) is in the domain of g. Identifying this domain ensures the composite function is defined and helps avoid invalid inputs.
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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)

Textbook Question

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

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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

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

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

Graph each function. f(x)={∣x∣if x<36−xif x≥3f(x) =\(\begin{cases}\)|x| & \(\text{if }\) x < 3 \\6 - x & \(\text{if }\) x \(\geq\) 3\(\end{cases}\)

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