For the real-valued functions \( f(x)=x^{2}-3 \) and \( g(x)=\sqrt{x+4} \), find the composition \( f \circ g \) and specify its \[ (f \circ g)(x)= \] Domain of \( f 0 g \) : \( \square \)

To find the composition \( f \circ g \), we need to substitute \( g(x) \) into \( f(x) \). Starting with: \[ g(x) = \sqrt{x + 4} \] we substitute this into \( f(x) \): \[ f(g(x)) = f(\sqrt{x + 4}) = (\sqrt{x + 4})^2 - 3. \] This simplifies to: \[ f(g(x)) = x + 4 - 3 = x + 1. \] Thus, \[ (f \circ g)(x) = x + 1. \] Now, regarding the domain of \( f \circ g \), we need to ensure that \( g(x) \) is defined and that \( f(g(x)) \) will also be valid. The function \( g(x) = \sqrt{x + 4} \) is defined for \( x + 4 \geq 0 \), which simplifies to \( x \geq -4 \). Since \( f(x) = x^2 - 3 \) is defined for all real numbers, the domain of \( f \circ g \) is the same as that of \( g(x) \). Therefore, the domain of \( f \circ g \) is: \[ \text{Domain of } (f \circ g) : [-4, \infty). \]