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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 21c
Chapter 3, Problem 21c

For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2.
ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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1
Identify the given functions: \( f(x) = 2x^2 - 3x \) and \( g(x) = x^2 - x + 3 \).
Find the composite function \( (f \circ g)(x) \), which means \( f(g(x)) \). This involves substituting \( g(x) \) into every \( x \) in \( f(x) \).
Write the expression for \( f(g(x)) \) by replacing \( x \) in \( f(x) \) with \( g(x) \): \[ f(g(x)) = 2(g(x))^2 - 3(g(x)) \].
Simplify the expression by first squaring \( g(x) = x^2 - x + 3 \), then multiply by 2, and subtract 3 times \( g(x) \).
Determine the domain of each function: since both \( f(x) \) and \( g(x) \) are polynomials, their domains are all real numbers. For the composite \( (f \circ g)(x) \), the domain is all real numbers as well.

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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 (ƒg)(x) = ƒ(g(x)). This means you first evaluate g(x), then substitute that output into ƒ. Understanding this process is essential to correctly find the composite function.
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Function Composition

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of (ƒg)(x) depends on the domain of g and the domain of ƒ evaluated at g(x). Identifying these domains ensures the composite function is valid.
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Domain Restrictions of Composed Functions

Polynomial Functions

Both ƒ(x) and g(x) are polynomial functions, which are defined for all real numbers. Recognizing this helps simplify domain considerations, as polynomials have no restrictions like division by zero or square roots of negative numbers.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

Graph each piecewise-defined function.

f(x)={6−xif x≤33if x>3f(x) =\(\begin{cases}\)6 - x & \(\text{if }\) x \(\leq\) 3 \\3 & \(\text{if }\) x > 3\(\end{cases}\)

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32.

x-intercept (3,0), y-intercept (0,-2)

Textbook Question

For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2.

ƒ(x)=2x^2-3x, g(x)=x^2-x+3

1
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Textbook Question

For the pair of functions defined, find (ƒ+g)(x).Give the domain of each. See Example 2.

ƒ(x)=2x^2-3x, g(x)=x^2-x+3

3
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Textbook Question

For the pair of functions defined, find (ƒ-g)(x). Give the domain of each. See Example 2.

ƒ(x)=2x^2-3x, g(x)=x^2-x+3

2
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