Find the Composite Function

For f(x)=
√
x
and g(x)=3x+9, find the following composite function.

Simplify your answer.

(f∘g)(x)=

Question

Find the Composite Function

For f(x)=
√
x
 and g(x)=3x+9, find the following composite function.

Simplify your answer.

(f∘g)(x)=

BlackTom AI · Accepted Answer

We are asked to find the composite function (f ∘ g)(x) where f(x) = √x and g(x) = 3x + 9.

Step 1: Apply g to x inside f. By definition of composition, (f ∘ g)(x) = f(g(x)).

Step 2: Substitute g(x) into f. Since f(t) = √t, with t = g(x) = 3x + 9, we get (f ∘ g)(x) = √(3x + 9).

Step 3: Consider domain for the composite function. The square root requires its argument to be nonnegative: 3x + 9 ≥ 0, which simplifies to x ≥ -3. Thus, the domain of (f ∘ g)(x) is x ≥ -3.

Step 4: Check possible simplifications. 3x + 9 factors as 3(x + 3), so √(3x + 9) can be written as √[3(x + 3)]. Depending on instructions, you might also express it as √3 · √(x + 3). However, √(3x + 9) is already a valid and compact form, and √3 · √(x + 3) is an alternative form that makes the factor 3 explicit.

Note on potential distractors (if options were present): any choice that misapplies the square root to either g(x) or f(x) or ignores the nonnegativity requirement would be incorrect. For example, selecting 3√(x + 3) would be incorrect because the square root should apply to the whole expression (3x + 9), not just to a part of it, and 3√(x + 3) would imply taking the square root of x+3 first and then multiplying by 3, which is not equivalent to √(3x + 9).

MAT-265-OD24-02-7799

Find the Composite Function For f(x)= √ x and g(x)=3x+9, find the following composite function. Simplify your answer. (f∘g)(x)=

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