Skip to main content
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 83a
Chapter 3, Problem 83a

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

Verified step by step guidance
1
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) = \(\sqrt{x+2}\), replace x with g(x) = -\(\frac{1}{x}\) to get (ƒ∘g)(x) = \(\sqrt{-\frac{1}{x}\) + 2}.
Step 3: Determine the domain of (ƒ∘g)(x). The domain must satisfy two conditions: (a) the domain of g(x), which excludes x = 0 because of division by zero, and (b) the expression inside the square root in ƒ(g(x)) must be greater than or equal to zero, so \(\left\)(-\(\frac{1}{x}\) + 2\(\right\)) \(\geq\) 0.
Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = -\(\frac{1}{x}\), replace x with ƒ(x) = \(\sqrt{x+2}\) to get (g∘ƒ)(x) = -\(\frac{1}{\sqrt{x+2}\)}.
Step 5: Determine the domain of (g∘ƒ)(x). The domain must satisfy: (a) the domain of ƒ(x), which requires the radicand \(\left\)(x+2\(\right\)) \(\geq\) 0, and (b) the denominator in g(ƒ(x)) cannot be zero, so \(\sqrt{x+2}\) \(\neq\) 0, which means \(\left\)(x+2\(\right\)) > 0.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

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.
Recommended video:
4:56
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 domain of the inner function and the domain restrictions of the outer function after substitution.
Recommended video:
3:51
Domain Restrictions of Composed Functions

Square Root and Rational Function Restrictions

The square root function requires the radicand to be non-negative, limiting inputs to values where x+2 ≥ 0. The rational function 1/x is undefined at x=0, so its domain excludes zero. These restrictions must be considered when finding domains of composed functions.
Recommended video:
05:21
Restrictions on Rational Equations
Related Practice
Textbook Question

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

798
views
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

44
views
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

1406
views
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)=-(\(\frac{2}{x}\))

29
views
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)=-(\(\frac{1}{x}\))

937
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

2094
views