Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 76

Chapter 3, Problem 76

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

Verified step by step guidance

Identify the given functions: $f(x) = \sqrt{x}$ and $g(x) = x - 1$.

For part (a), find the composition $(f \circ g)(x)$, which means $f(g(x))$. Substitute $g(x)$ into $f$: write $f(g(x)) = f(x - 1) = \sqrt{x - 1}$.

Determine the domain of $(f \circ g)(x)$. Since $f(x) = \sqrt{x}$ requires the input to be greater than or equal to zero, set the inside of the square root $x - 1 \geq 0$ and solve for $x$.

For part (b), find the composition $(g \circ f)(x)$, which means $g(f(x))$. Substitute $f(x)$ into $g$: write $g(f(x)) = g(\sqrt{x}) = \sqrt{x} - 1$.

Determine the domain of $(g \circ f)(x)$. Since $f(x) = \sqrt{x}$ requires $x \geq 0$, and $g$ is defined for all real numbers, the domain of $(g \circ f)(x)$ is 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 (f∘g)(x) = f(g(x)). It requires substituting the entire output of g(x) into the function f. Understanding this process is essential to correctly form the composite functions in the problem.

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 the composite function depends on the domains of both functions and the values that keep the inner function's output within the outer function's domain.

Square Root Function Domain Restrictions

The square root function ƒ(x) = √x is only defined for x ≥ 0 because the square root of a negative number is not a real number. This restriction affects the domain of composite functions involving ƒ, requiring careful consideration of the inner function's output to ensure non-negativity.

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Domain Restrictions of Composed Functions

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

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

$ƒ (x) = x^{3}, g (x) = x^{2} + 3 x - 1$

Textbook Question

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

$ƒ (x) = √ x, g (x) = x + 3$

Textbook Question

Graph each function. $f (x) = {\begin{matrix} ∣ x ∣ & if x < 3 \\ 6 - x & if x \geq 3 \end{matrix}$

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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 $y = x^{2}$ ?