Specify the domain of $( g \circ f ) ( x )$ for the pair of functions $f ( x ) = \sqrt { x + 9 } \text { and } g ( x ) = 5 x + 9$
A) $( - \infty , 9 ]$
B) $[ 0 , \infty )$
C) $( - \infty , \infty )$
D) $[ - 9 , \infty )$

Question

Quizplus · Accepted Answer

To find the domain of $( g \circ f ) ( x )$, we need to first find the range of the function $f(x)$. Since $f(x) = \sqrt{x + 9}$ and the radicand must be non-negative, we have $x + 9 \geq 0$, which means the domain of $f(x)$ is $[-9, \infty)$.

Next, we need to find the domain of $(g \circ f)(x)$, which is the composition of $g(x)$ and $f(x)$. We have:

$$(g \circ f)(x) = g(f(x)) = g(\sqrt{x+9}) = 5\sqrt{x+9} + 9$$

Since the square root must be non-negative, the domain of $(g \circ f)(x)$ is the same as the domain of $f(x)$, which is $[ - 9 , \infty )$. Therefore, the answer is D.

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