State the Domain

For f(x)=
√
x
and g(x)=3x+9,

State the domain of (f∘g)(x)=

Simply your answer.

Question

State the Domain

For f(x)=
√
x
 and g(x)=3x+9,

State the domain of (f∘g)(x)=

Simply your answer.

BlackTom AI · Accepted Answer

To determine the domain of the composition (f ∘ g)(x) = f(g(x)) when f(x) = √x and g(x) = 3x + 9, we follow these steps.

First, examine the inner function: g(x) = 3x + 9 is defined for all real numbers, since it is a linear function with no restrictions.

Next, apply the outer function: f(u) = √u requires its input u to be nonnegative, i.e., u ≥ 0, because the square root is defined only for nonnegative arguments in the real numbers.

Here, the input to f is g(x) = 3x + 9. Therefore we require 3x + 9 ≥ 0. Solving this inequality gives x ≥ -3.

Thus, the domain of (f ∘ g)(x) consists of all real x such that x ≥ -3. If you compare alternative candidate domains, you should verify whether they satisfy the same inequality 3x + 9 ≥ 0; any domain that includes x < -3 would be invalid because it would produce a negative input under the square root.

In summary, the only necessary restriction comes from the square root: x must be at least -3. Any proposed domain not obeying x ≥ -3 would be incorrect.

MAT-265-OD24-02-7799

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

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