Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=−6x+9,g(x)=5x+7ƒ(x)=-6x+9, g(x)=5x+7

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 73a

Chapter 3, Problem 73a

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = - 6 x + 9, g (x) = 5 x + 7$

Verified step by step guidance

Identify the given functions: $f(x) = -6x + 9$ and $g(x) = 5x + 7$.

Recall that the composition $(f \circ g)(x)$ 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 write $f(g(x)) = -6(g(x)) + 9$.

Simplify the expression by distributing and combining like terms: $f(g(x)) = -6(5x + 7) + 9$.

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 combine the two given functions.

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 find valid inputs. Identifying these restrictions ensures the composite function is well-defined.

Linear Functions and Their Properties

Both f(x) = -6x + 9 and g(x) = 5x + 7 are linear functions, which are defined for all real numbers. Knowing that linear functions have no domain restrictions simplifies finding the domain of the composite function, as the domain will typically be all real numbers unless otherwise specified.

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