Math Snap

STEP 1

1. The composite function $g \circ f$ is defined as $g(f(x))$.
2. The domain of a composite function $g \circ f$ consists of all $x$ such that $f(x)$ is in the domain of $g$.

STEP 2

1. Determine the expression for $g \circ f$.
2. Determine the domain of $g \circ f$.

STEP 3

First, find the expression for $g \circ f$. This means substituting $f(x)$ into $g(x)$:
Given:
$$f(x) = x^2 + 4$$ $$g(x) = \sqrt{x}$$ The composite function $g \circ f$ is:
$$g(f(x)) = g(x^2 + 4) = \sqrt{x^2 + 4}$$

Determine the domain of $g \circ f$.
The function $g(x) = \sqrt{x}$ is defined for $x \geq 0$. Therefore, $f(x)$ must satisfy:
$$x^2 + 4 \geq 0$$ Since $x^2 + 4$ is always positive for all real numbers $x$ (as $x^2 \geq 0$ and adding 4 makes it strictly positive), the domain of $g \circ f$ is all real numbers.
The correct answer is:
c $\sqrt{x^{2}+4}$ with Domain of $g \circ f$: all real numbers $x$.