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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 85b
Chapter 3, Problem 85b

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)=\(\frac{1}{(x+5)}\)

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First, understand that the composition of functions (g \(\circ\) f)(x) means you substitute the function f(x) into g(x). In other words, (g \(\circ\) f)(x) = g(f(x)).
Given f(x) = \(\sqrt{x}\) and g(x) = \(\frac{1}{x+5}\), substitute f(x) into g(x) to get (g \(\circ\) f)(x) = \(\frac{1}{\sqrt{x}\) + 5}.
Next, determine the domain of (g \(\circ\) f)(x). Start by considering the domain of f(x) = \(\sqrt{x}\), which requires that the expression inside the square root is non-negative, so x \(\geq\) 0.
Then, consider the domain restrictions from g(x) when applied to f(x). Since g(x) = \(\frac{1}{x+5}\), the denominator cannot be zero. So, set \(\sqrt{x}\) + 5 \(\neq\) 0 and solve for x.
Combine the domain restrictions from both functions to find the overall domain of (g \(\circ\) f)(x). This means x must satisfy both x \(\geq\) 0 and \(\sqrt{x}\) + 5 \(\neq\) 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 requires substituting the entire output of f(x) into g(x), creating a new function that combines both operations.
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Function Composition

Domain of a Function

The domain is the set of all input values for which a function is defined. When composing functions, the domain of (g∘f)(x) includes all x-values in the domain of f for which f(x) is in the domain of g.
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Domain Restrictions of Composed Functions

Square Root and Rational Function Domains

The square root function ƒ(x) = √x requires x ≥ 0 to avoid imaginary numbers. The rational function g(x) = 1/(x+5) is undefined when the denominator is zero, so x ≠ -5. These restrictions affect the domain of the composite function.
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Imaginary Roots with the Square Root Property
Related Practice
Textbook Question

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)=\(\frac{1}{(x+5)}\)

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

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. 5y2+5x2=305y^2 + 5x^2 =30

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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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