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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 114
Chapter 3, Problem 114

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)

Verified step by step guidance
1
Understand that the composition of functions (ƒ ○ g)(x) means ƒ(g(x)), which is the function ƒ applied to the output of g(x).
Identify the given functions: ƒ(x) = \(\sqrt{x - 2}\) and g(x) = x^2.
Substitute g(x) into ƒ(x) to get (ƒ ○ g)(x) = ƒ(g(x)) = \(\sqrt{g(x) - 2}\).
Replace g(x) with x^2 inside the square root to write (ƒ ○ g)(x) = \(\sqrt{x^2 - 2}\).
Consider the domain restrictions: since the expression inside the square root must be greater than or equal to zero, set up the inequality x^2 - 2 \(\geq\) 0 and solve for x to find the domain of (ƒ ○ g)(x).

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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)). It means you first evaluate g(x), then use that output as the input for ƒ. Understanding this process is essential to correctly combine functions.
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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 the composite depends on the domain of the inner function and the domain restrictions of the outer function applied to the inner function's output.
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Domain Restrictions of Composed Functions

Square Root Function Restrictions

The square root function √(x) is only defined for x ≥ 0 in the real numbers. When composing with other functions, ensure the expression inside the square root is non-negative to find the valid domain and avoid undefined values.
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Imaginary Roots with the Square Root Property
Related Practice
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)

Textbook Question

Let ƒ(x)=3x2−4ƒ(x) = 3x^2 - 4 and g(x)=x2−3x−4g(x) = x^2 - 3x -4. Find each of the following.

(f/g)(−1)(f/g)(-1)

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. the domain of ƒ○gƒ ○ g

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.

(f○g)(−6)(f ○ g)(-6)

Textbook Question

Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

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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)(x)(g ○ ƒ)(x)