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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 19c
Chapter 3, Problem 19c

For the pair of functions defined, find (ƒg)(x).Give the domain of each. See Example 2.
ƒ(x)=3x+4, g(x)=2x-7

Verified step by step guidance
1
Understand that (ƒg)(x) means the composition of functions, which is ƒ(g(x)). This means you will substitute the entire function g(x) into the function ƒ(x).
Write down the given functions: ƒ(x) = 3x + 4 and g(x) = 2x - 7.
Substitute g(x) into ƒ(x): replace every x in ƒ(x) with g(x), so (ƒg)(x) = 3(2x - 7) + 4.
Simplify the expression by distributing and combining like terms: multiply 3 by each term inside the parentheses and then add 4.
Determine the domain of each function: since both ƒ(x) and g(x) are linear functions, their domains are all real numbers, but confirm this by considering any restrictions from the composition.

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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 result into ƒ(x). 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 the composite function depends on the domains of both functions and the outputs of the inner function.
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Domain Restrictions of Composed Functions

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They are defined for all real numbers, making their domain all real numbers. Recognizing this helps simplify finding domains in compositions involving linear functions.
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Related Practice
Textbook Question

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

ƒ(x)=3x+4, g(x)=2x-8

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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.

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)=3x+4, g(x)=2x-6

Textbook Question

For each piecewise-defined function, find (a) ƒ(-5), (b) ƒ(-1), (c) ƒ(0), and (d) ƒ(3).

f(x)={2+xif x<−4−xif −4≤x≤23xif x>2f(x) =\(\begin{cases}\)2 + x & \(\text{if }\) x < -4 \\-x & \(\text{if }\) -4 \(\leq\) x \(\leq\) 2 \\3x & \(\text{if }\) x > 2\(\end{cases}\)

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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)=3x+4, g(x)=2x-5

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Textbook Question

Determine whether each relation defines a function, and give the domain and range. {(2,5),(3,7),(3,9),(5,11)}