Determining Composite Functions and Their Domains Given that ƒ(x) [latex]= \sqrt{x}[/latex] and g(x) = 4x + 2, find each of the following. (a) (ƒ ∘ g)(x) and its domain (b) (g ∘ ƒ)(x) and its domain

(a) (ƒ ∘ g)(x) = ƒ(g(x)) Definition of composition = ƒ(4x + 2) g(x) = 4x + 2 [latex]= \sqrt{4x + 2} ƒ(x)= \sqrt{x}[/latex] The domain and range of g are both the set of all real numbers, (-∞, ∞). The domain of ƒ is the set of all nonnegative real numbers, [0, ∞). Thus, g(x), which is defined as 4x + 2, must be greater than or equal to zero. 4x + 2 ≥ 0 Solve the inequality. [latex]x ≥ -\frac{1}{2}[/latex] Subtract 2. Divide by 4. Therefore, the domain of ƒ ∘ g is [[latex] -\frac{1}{2} , ∞[/latex]). (b) (g ∘ ƒ)(x) = g(ƒ(x)) Definition of composition [latex]= g(\sqrt{x}) ƒ(x) =\sqrt{x}[/latex] [latex]= 4\sqrt{x} + 2[/latex] g(x)= 4x + 2 The domain and range of ƒ are both the set of all nonnegative real numbers, [0, ∞). The domain of g is the set of all real numbers, (-∞, ∞). Therefore, the domain of g ∘ ƒ is [0, ∞).

Question 2.8.6: Determining Composite Functions and Their Domains Given that...

Precalculus

Determining Composite Functions and Their Domains

Given that ƒ(x) $= \sqrt{x}$ and g(x) = 4x + 2, find each of the following.

(a) (ƒ ∘ g)(x) and its domain (b) (g ∘ ƒ)(x) and its domain

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

(a) (ƒ ∘ g)(x)

= ƒ(g(x)) Definition of composition

= ƒ(4x + 2) g(x) = 4x + 2

$= \sqrt{4x + 2} ƒ(x)= \sqrt{x}$

The domain and range of g are both the set of all real numbers, (-∞, ∞). The domain of ƒ is the set of all nonnegative real numbers, [0, ∞). Thus, g(x), which is defined as 4x + 2, must be greater than or equal to zero.

4x + 2 ≥ 0 Solve the inequality.

$x ≥ -\frac{1}{2}$ Subtract 2. Divide by 4.

Therefore, the domain of ƒ ∘ g is [ $-\frac{1}{2} , ∞$ ).

(b) (g ∘ ƒ)(x)

= g(ƒ(x)) Definition of composition

$= g(\sqrt{x}) ƒ(x) =\sqrt{x}$

$= 4\sqrt{x} + 2$ g(x)= 4x + 2

The domain and range of ƒ are both the set of all nonnegative real numbers, [0, ∞). The domain of g is the set of all real numbers, (-∞, ∞). Therefore, the domain of g ∘ ƒ is [0, ∞).