Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=√x,g(x)=x+3ƒ(x)=√x, g(x)=x+3

Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 75a

Chapter 3, Problem 75a

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = √ x, g (x) = x + 3$

Verified step by step guidance

Identify the given functions: $f(x) = \sqrt{x}$ and $g(x) = x + 3$.

Find the composition $(f \circ g)(x)$, which means $f(g(x))$. Substitute $g(x)$ into $f$: $(f \circ g)(x) = f(x + 3) = \sqrt{x + 3}$.

Determine the domain of $(f \circ g)(x)$ by considering the domain restrictions of the square root function. Since the expression inside the square root must be non-negative, set $x + 3 \geq 0$.

Solve the inequality $x + 3 \geq 0$ to find the domain: $x \geq -3$.

Express the domain of $(f \circ g)(x)$ as all real numbers $x$ such that $x \geq -3$.

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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 means you first evaluate g(x), then use that output as the input for f. Understanding this process is essential to correctly find (f∘g)(x).

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) depends on the domain of g and the domain of f applied to g(x). Identifying domain restrictions ensures the composition is valid.

Square Root Function and Its Domain

The square root function, f(x) = √x, is defined only for x ≥ 0 because the square root of a negative number is not real. When composing with g(x), you must ensure that g(x) ≥ 0 to keep the composition within the real numbers.

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Related Practice

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Textbook Question

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