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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 85a
Chapter 3, Problem 85a

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.
ƒ(x)=√x,g(x)=1(x+5)ƒ(x)=\(\surd\) x,g(x)=\(\frac{1}{(x+5)}\)

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Identify the given functions: \(f(x) = \sqrt{x}\) and \(g(x) = \frac{1}{x+5}\).
Find the composition \((f \circ g)(x)\), which means substituting \(g(x)\) into \(f\). So, write \((f \circ g)(x) = f(g(x)) = \sqrt{g(x)}\).
Substitute \(g(x)\) into \(f\): \((f \circ g)(x) = \sqrt{\frac{1}{x+5}}\).
Determine the domain of \((f \circ g)(x)\) by considering the restrictions from both \(f\) and \(g\). Since \(f(x) = \sqrt{x}\) requires the input to be \(\geq 0\), set the inside of the square root \(\frac{1}{x+5} \geq 0\).
Solve the inequality \(\frac{1}{x+5} \geq 0\) to find the domain of \((f \circ g)(x)\). Also, exclude any values that make the denominator zero, i.e., \(x \neq -5\).

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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 function.
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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 (f∘g)(x) depends on the domain of g and the values of g(x) that lie within the domain of f. Identifying these restrictions ensures the composite function is valid.
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Domain Restrictions of Composed Functions

Square Root and Rational Function Restrictions

The square root function requires its input to be non-negative, so the expression inside must be ≥ 0. The rational function 1/(x+5) is undefined when the denominator is zero, so x ≠ -5. These restrictions must be considered when finding the domain of the composite function.
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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)=√x,g(x)=1(x+5)ƒ(x)=\(\surd\) x,g(x)=\(\frac{1}{(x+5)}\)

Textbook Question

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

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

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

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

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