Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(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

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 75b

Chapter 3, Problem 75b

Given functions f and g, find (b) $(g \circ ƒ) (x)$ 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 $.

Recall that the composition $ (g \circ f)(x) $ means $ g(f(x)) $, which is applying $ f $ first, then $ g $ to the result.

Substitute $ f(x) $ into $ g $: $ (g \circ f)(x) = g(\sqrt{x}) = \sqrt{x} + 3 $.

Determine the domain of $ (g \circ f)(x) $ by considering the domain of $ f(x) $ first, since $ f $ is applied first. Since $ f(x) = \sqrt{x} $, the domain is $ x \geq 0 $ because the square root requires non-negative inputs.

Since $ g(x) = x + 3 $ is defined for all real numbers, the domain of $ (g \circ f)(x) $ is the same as the domain of $ f(x) $, which is $ x \geq 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 means you first evaluate f at x, then use that output as the input for g. Understanding this process is essential to correctly find (g∘f)(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 (g∘f) depends on the domain of f and the domain restrictions of g applied to f(x). Identifying these restrictions ensures the composition is valid.

Square Root Function Domain

The square root function ƒ(x) = √x is defined only for x ≥ 0 because the square root of a negative number is not a real number. This domain restriction affects the composition since f(x) must produce values within g's domain, and x must satisfy x ≥ 0.

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Related Practice

Textbook Question

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. h(x)=-(x+1)³

Textbook Question

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

$ƒ (x) = √ x, g (x) = x + 3$

Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

$y = ∣ x - 1 ∣$

$y = x^{2} - 4$

How is the graph of y = x^2 - 4 obtained by transforming the graph of $y = x^{2}$ ?

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)=x-1