Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=8x+12,g(x)=3x−1ƒ(x)=8x+12, g(x)=3x-1

Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 74a

Chapter 3, Problem 74a

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = 8 x + 12, g (x) = 3 x - 1$

Verified step by step guidance

Identify the given functions: $f(x) = 8x + 12$ and $g(x) = 3x - 1$.

Recall that the composition of functions $(f \circ g)(x)$ means $f(g(x))$, which is the function $f$ evaluated at $g(x)$.

Substitute $g(x)$ into $f(x)$: replace every $x$ in $f(x)$ with $g(x)$, so $(f \circ g)(x) = f(3x - 1) = 8(3x - 1) + 12$.

Simplify the expression by distributing and combining like terms to write $(f \circ g)(x)$ in standard form.

Determine the domain of $(f \circ g)(x)$ by considering the domain of $g(x)$ and the domain of $f$ evaluated at $g(x)$. Since both $f$ and $g$ are linear functions, 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).

Domain of a Composite Function

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. This means you must consider restrictions from both functions to determine where the composition is defined.

Linear Functions

Both f(x) = 8x + 12 and g(x) = 3x - 1 are linear functions, which are defined for all real numbers. Recognizing this helps simplify finding the domain of the composite function, as there are no domain restrictions from either function.

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Related Practice

Textbook Question

Given functions f and g, find (b) $(g \circ ƒ) (x)$ and its domain. See Examples 6 and 7.

$ƒ (x) = 8 x + 12, g (x) = 3 x - 1$

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, 4), (-2, -1), (1, 14)

Textbook Question

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

$ƒ (x) = - 6 x + 9, g (x) = 5 x + 7$

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. h(x)=-(x+1)³

Textbook Question

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

$ƒ (x) = - 6 x + 9, g (x) = 5 x + 7$

Textbook Question

Graph each function. ƒ(x) = 2∛(x+1)-2

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