Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(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

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 74b

Chapter 3, Problem 74b

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

Verified step by step guidance

Recall that the composition of functions $ (g \circ f)(x) $ means you substitute $ f(x) $ into $ g(x) $. In other words, $ (g \circ f)(x) = g(f(x)) $.

Given $ f(x) = 8x + 12 $ and $ g(x) = 3x - 1 $, substitute $ f(x) $ into $ g $ to get $ g(f(x)) = 3(8x + 12) - 1 $.

Simplify the expression by distributing the 3 across $ 8x + 12 $, which gives $ 3 \times 8x + 3 \times 12 - 1 $.

Combine like terms to write the simplified form of $ (g \circ f)(x) $.

To find the domain of $ (g \circ f)(x) $, consider the domain of $ f(x) $ first, then ensure that the output of $ f(x) $ fits within the domain of $ g(x) $. Since both $ f $ and $ g $ are linear functions, 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 find (g∘f)(x).

Evaluating Linear Functions

Both f(x) = 8x + 12 and g(x) = 3x - 1 are linear functions. Evaluating these functions means substituting a value for x and simplifying. Knowing how to substitute and simplify expressions is key to finding the composite function.

Domain of Composite Functions

The domain of (g∘f)(x) consists of all x-values in the domain of f for which f(x) is in the domain of g. Since both f and g are linear with domains of all real numbers, the composite function's domain is also all real numbers, but this concept is crucial for more complex functions.

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Related Practice

Textbook Question

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

$ƒ (x) = 8 x + 12, g (x) = 3 x - 1$

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

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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Textbook Question

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

$ƒ (x) = √ x, g (x) = x + 3$