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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 82
Chapter 3, Problem 82

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
ƒ(x)=4x,g(x)=x+4ƒ(x)=\(\frac{4}{x}\),g(x)=x+4

Verified step by step guidance
1
Step 1: Understand the composition of functions. The composition (ƒ∘g)(x) means ƒ(g(x)), which is applying g first, then ƒ to the result. Similarly, (g∘ƒ)(x) means g(ƒ(x)).
Step 2: Find (ƒ∘g)(x) by substituting g(x) into ƒ. Since ƒ(x) = 4/x and g(x) = x + 4, write (ƒ∘g)(x) = ƒ(g(x)) = 4 / (x + 4).
Step 3: Determine the domain of (ƒ∘g)(x). The domain consists of all x-values for which the expression is defined. Since the denominator cannot be zero, set x + 4 ≠ 0 and solve for x.
Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = x + 4 and ƒ(x) = 4/x, write (g∘ƒ)(x) = g(ƒ(x)) = (4/x) + 4.
Step 5: Determine the domain of (g∘ƒ)(x). The domain consists of all x-values for which ƒ(x) is defined (since it is inside g). Since ƒ(x) = 4/x, x cannot be zero.

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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 requires substituting the entire output of g(x) into the function f. Understanding this process is essential to correctly form the composite functions in the problem.
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Function Composition

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When composing functions, the domain of the composite function depends on the domains of both functions and the values for which the inner function's output lies within the domain of the outer function.
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Domain Restrictions of Composed Functions

Rational Functions and Restrictions

A rational function is a ratio of polynomials, such as f(x) = 4/x, which is undefined when the denominator is zero. Identifying values that make the denominator zero is crucial to determine domain restrictions, especially when these functions are part of compositions.
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Restrictions on Rational Equations
Related Practice
Textbook Question

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

ƒ(x)=2x,g(x)=x+1ƒ(x)=\(\frac{2}{x}\),g(x)=x+1

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Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=√(x+4),g(x)=−(2x)ƒ(x)=\(\surd\)(x+4),g(x)=-(\(\frac{2}{x}\))

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Textbook Question

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

ƒ(x)=√(x+2),g(x)=−(1x)ƒ(x)=\(\surd\)(x+2),g(x)=-(\(\frac{1}{x}\))

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Textbook Question

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

ƒ(x)=√(x+2),g(x)=−(1x)ƒ(x)=\(\surd\)(x+2),g(x)=-(\(\frac{1}{x}\))

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Textbook Question

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

ƒ(x)=2x,g(x)=x+1ƒ(x)=\(\frac{2}{x}\),g(x)=x+1

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Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√(x-2), g(x)=2x

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