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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 88
Chapter 3, Problem 88

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=1/(x+4), g(x)=-(1/x)

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Step 1: Understand the composition of functions. For (ƒ∘g)(x), this means ƒ(g(x)), which is the function ƒ applied to the output of g(x). Similarly, for (g∘ƒ)(x), this means g(ƒ(x)).
Step 2: Find (ƒ∘g)(x) by substituting g(x) into ƒ. Since ƒ(x) = \(\frac{1}{x+4}\) and g(x) = -\(\frac{1}{x}\), replace x in ƒ(x) with g(x) to get ƒ(g(x)) = \(\frac{1}{\left(-\frac{1}{x}\)\(\right\)) + 4}.
Step 3: Determine the domain of (ƒ∘g)(x). Start with the domain of g(x), which excludes x = 0 because of division by zero. Then, ensure the expression inside ƒ, which is g(x), does not make the denominator zero in ƒ(g(x)). So solve \(\left\)(-\(\frac{1}{x}\)\(\right\)) + 4 \(\neq\) 0 to find any additional restrictions.
Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = -\(\frac{1}{x}\) and ƒ(x) = \(\frac{1}{x+4}\), replace x in g(x) with ƒ(x) to get g(ƒ(x)) = -\(\frac{1}{\frac{1}{x+4}\)}.
Step 5: Determine the domain of (g∘ƒ)(x). Start with the domain of ƒ(x), which excludes x = -4 due to division by zero. Then ensure the input to g, which is ƒ(x), does not cause division by zero in g(ƒ(x)). So find values of x for which \(\frac{1}{x+4}\) \(\neq\) 0 and also check for any other restrictions.

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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 f(x), creating a new function that combines both operations sequentially.
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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 any restrictions caused by operations like division by zero.
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Domain Restrictions of Composed Functions

Rational Functions and Restrictions

Rational functions are ratios of polynomials, often with restrictions where the denominator equals zero. Identifying these restrictions is crucial when finding domains, especially in compositions, to avoid undefined expressions such as division by zero.
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Related Practice
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