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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 78a
Chapter 3, Problem 78a

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

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1
Identify the given functions: \(f(x) = x + 2\) and \(g(x) = x^4 + x^2 - 4\).
To find the composition \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\), which means replacing every \(x\) in \(f(x)\) with \(g(x)\): \((f \circ g)(x) = f(g(x)) = g(x) + 2\).
Write the composed function explicitly by plugging in \(g(x)\): \((f \circ g)(x) = (x^4 + x^2 - 4) + 2\).
Simplify the expression inside the composition by combining like terms: \((f \circ g)(x) = x^4 + x^2 - 2\).
Determine the domain of \((f \circ g)(x)\) by considering the domain of \(g(x)\) first, then ensuring the output of \(g(x)\) fits into the domain of \(f\). 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(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

Both f(x) = x + 2 and g(x) = x^4 + x^2 - 4 are polynomial functions, which are expressions involving variables raised to whole-number powers with coefficients. Recognizing their polynomial nature helps in simplifying and evaluating compositions without domain restrictions from radicals or denominators.
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Introduction to Polynomial Functions

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 f and g are polynomials, their domains are all real numbers, but verifying this ensures the composite function is defined for all inputs.
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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

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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, (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

Textbook Question

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

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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