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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 81b
Chapter 3, Problem 81b

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

Verified step by step guidance
1
Recall that the composition of functions (g \(\circ\) f)(x) means g(f(x)), which is the function g applied to the output of f(x).
Start by substituting f(x) into g(x). Since f(x) = \(\frac{2}{x}\), replace the input of g with \(\frac{2}{x}\), so (g \(\circ\) f)(x) = g\(\left\)(\(\frac{2}{x}\)\(\right\)).
Now, apply the function g to \(\frac{2}{x}\). Since g(x) = x + 1, replace x in g(x) with \(\frac{2}{x}\), giving (g \(\circ\) f)(x) = \(\frac{2}{x}\) + 1.
Next, determine the domain of (g \(\circ\) f)(x). The domain consists of all x-values for which f(x) is defined and for which g(f(x)) is defined. Since f(x) = \(\frac{2}{x}\), x cannot be zero (division by zero is undefined). Also, check if g has any restrictions on its input; since g(x) = x + 1 is defined for all real numbers, no further restrictions come from g.
Therefore, the domain of (g \(\circ\) f)(x) is all real numbers except x = 0.

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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 at x, then use that output as the input for g. Understanding this process is essential to correctly find (g∘f)(x).
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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 (g∘f) includes all x-values in the domain of f for which f(x) is in the domain of g. Identifying domain restrictions is crucial to avoid undefined expressions.
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Domain Restrictions of Composed Functions

Rational Functions and Their Restrictions

A rational function is a ratio of two polynomials, like f(x) = 2/x, which is undefined where the denominator is zero. Recognizing these restrictions helps determine the domain of f and, consequently, the domain of the composition (g∘f).
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Restrictions on Rational Equations
Related Practice
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)=4x,g(x)=x+4ƒ(x)=\(\frac{4}{x}\),g(x)=x+4

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)=-(\(\frac{1}{x}\))

1
views
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)=-(\(\frac{1}{x}\))

1
views
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

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

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=-3(x-2)2+1