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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 118
Chapter 3, Problem 118

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible. the domain of ƒ○gƒ ○ g

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Recall that the composition of functions \( (f \circ g)(x) \) means \( f(g(x)) \). So here, \( (f \circ g)(x) = f(g(x)) = f(x^2) \).
Substitute \( g(x) = x^2 \) into \( f(x) = \sqrt{x - 2} \) to get \( f(g(x)) = \sqrt{x^2 - 2} \).
To find the domain of \( f \circ g \), determine for which values of \( x \) the expression inside the square root is non-negative, since the square root function requires the radicand to be \( \geq 0 \).
Set up the inequality: \( x^2 - 2 \geq 0 \). Solve this inequality to find the values of \( x \) that satisfy it.
Solve \( x^2 - 2 \geq 0 \) by isolating \( x^2 \): \( x^2 \geq 2 \). Then find the values of \( x \) such that \( x \leq -\sqrt{2} \) or \( x \geq \sqrt{2} \). These intervals form the domain of \( f \circ g \).

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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 (ƒ ○ g)(x) = ƒ(g(x)). To find the composition, you substitute g(x) into ƒ(x), which requires understanding how the output of g affects the input of ƒ.
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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 dealing with compositions, the domain of ƒ ○ g consists of all x-values in the domain of g such that g(x) lies within the domain of ƒ.
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Domain Restrictions of Composed Functions

Square Root Function Domain Restrictions

For a square root function like ƒ(x) = √(x-2), the expression inside the root must be non-negative. This means x - 2 ≥ 0, so the domain of ƒ is all x ≥ 2. When composing, this restriction applies to g(x), requiring g(x) ≥ 2.
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Related Practice
Textbook Question

Use the table to evaluate each expression, if possible.

(f−g)(3)(f-g)(3)

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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible

(g○ƒ)(3) (g ○ ƒ)(3)

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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.

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Textbook Question

Use the table to evaluate each expression, if possible.

(f/g)(0)(f/g) (0)

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Textbook Question

Use the table to evaluate each expression, if possible.

(ƒ+g)(1)(ƒ+g)(1)

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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.

(ƒ○g)(x)(ƒ ○ g)(x)

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