Given functions \( f \) and \( g \), find \( (a)(f \circ g)(x) \) and its domain, and \( (b)(g \circ f)(x) \) and its domain. \[ f(x)=\sqrt{x}, g(x)=x+7 \] (a) \( (f \circ g)(x)=\square \) (Simplify your answer. Type an exact answer, using radicals as needed.) The domain of \( f \circ g \) is \( \square \). (Type your answer in interval notation.) (b) (gof)(x) \( =\square \) (Simplify your answer. Type an exact answers, using radicals as needed.) The domain of \( g \) of is \( \square \). (Type your answer in interval notation.)

To find \( (a)(f \circ g)(x) \), we start by substituting \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(x + 7) = \sqrt{x + 7} \] So, \( (f \circ g)(x) = \sqrt{x + 7} \). Now, let's determine the domain. Since we have the square root function, we need \( x + 7 \geq 0 \): \[ x \geq -7 \] Therefore, the domain of \( f \circ g \) is \( [-7, \infty) \). (b) Now for \( (g \circ f)(x) \), we substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 7 \] So, \( (g \circ f)(x) = \sqrt{x} + 7 \). Next, we need to find the domain of \( g \) of \( f \). The function \( f(x) = \sqrt{x} \) requires that \( x \geq 0 \): \[ x \geq 0 \] Thus, the domain of \( g \circ f \) is \( [0, \infty) \).