State the Domain

For
𝑓
(
𝑥
)
=
𝑥
and
𝑔
(
𝑥
)
=
3
𝑥
+
9
,

State the domain of
(
𝑓
∘
𝑔
)
(
𝑥
)
=

Simply your answer.

Question

State the Domain

For 
𝑓
(
𝑥
)
=
𝑥
 and 
𝑔
(
𝑥
)
=
3
𝑥
+
9
,

State the domain of 
(
𝑓
∘
𝑔
)
(
𝑥
)
=

Simply your answer.

BlackTom AI · Accepted Answer

We start by parsing the problem: f(x) = x (the identity function) and g(x) = 3x + 9. We are asked for the domain of the composition (f ∘ g)(x) = f(g(x)).

First, compute the composition: f(g(x)) means we plug g(x) into f. Since f(t) = t for any input t, f(g(x)) = g(x) = 3x + 9.

Next, determine the domain of the resulting expression. The function 3x + 9 is a polynomial, and polynomials are defined for all real x. Therefore, the domain is all real numbers.

Now evaluate each option:
- Option 1: { x | x ≥ −9/3 } simplifies to { x | x ≥ −3 }. This restricts x to x ≥ −3, which is too limited because the domain actually includes all real numbers, not just those greater than or equal to −3.
- Option 2: { x | x ≥ 0 }. This states a half-line starting at 0, again narrower than the actual domain.
- Option 3: { x | x ≤ 0 }. This provides the non-positive half-line, which is also incorrect.
- Option 4: all real numbers. This matches the true domain of the polynomial 3x + 9.

In summary, since the composition reduces to a linear polynomial with no domain restrictions, any answer that limits x (to ≥ −3, ≥ 0, or ≤ 0) is incorrect. The only correct characterization is that the domain is all real numbers.

MAT-265-OD24-02-7799 Prior Knowledge Check

State the Domain For 𝑓 ( 𝑥 ) = 𝑥 and 𝑔 ( 𝑥 ) = 3 𝑥 + 9 , State the domain of ( 𝑓 ∘ 𝑔 ) ( 𝑥 ) = Simply your answer.

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