For \( f(x)=\sqrt{x} \) and \( g(x)=x+6 \), find the following functions. a. \( (f \circ g)(x) ; b .(g \circ f)(x) ; \) c. \( (f \circ g)(3) \); d. \( (g \circ f)(3) \) a. \( (f \circ g)(x)=\sqrt{x+6} \) (Simplify your answer.) b. \( (g \circ f)(x)=\sqrt{x}+6 \) (Simplify your answer.) c. \( (f \circ g)(3)=\square \) (Simplify your answer.)

Evaluate the expression by following steps: - step0: Evaluate: \(\sqrt{x}+6\) - step1: Substitute: \(\sqrt{3}+6\) Substitute \( x=3 \) into the expression \( \sqrt{x+6} \). Evaluate the expression by following steps: - step0: Evaluate: \(\sqrt{x+6}\) - step1: Substitute: \(\sqrt{3+6}\) - step2: Add the numbers: \(\sqrt{9}\) - step3: Write in exponential form: \(\sqrt{3^{2}}\) - step4: Simplify the root: \(3\) Let's go through the functions step by step: ### a. \( (f \circ g)(x) \) The composition of functions \( f \) and \( g \) is defined as: \[ (f \circ g)(x) = f(g(x)) \] Substituting \( g(x) = x + 6 \) into \( f(x) = \sqrt{x} \): \[ (f \circ g)(x) = f(x + 6) = \sqrt{x + 6} \] So, the simplified answer is: \[ (f \circ g)(x) = \sqrt{x + 6} \] ### b. \( (g \circ f)(x) \) The composition of functions \( g \) and \( f \) is defined as: \[ (g \circ f)(x) = g(f(x)) \] Substituting \( f(x) = \sqrt{x} \) into \( g(x) = x + 6 \): \[ (g \circ f)(x) = g(\sqrt{x}) = \sqrt{x} + 6 \] So, the simplified answer is: \[ (g \circ f)(x) = \sqrt{x} + 6 \] ### c. \( (f \circ g)(3) \) Now, we need to evaluate \( (f \circ g)(3) \): \[ (f \circ g)(3) = \sqrt{3 + 6} = \sqrt{9} = 3 \] So, the answer is: \[ (f \circ g)(3) = 3 \] ### d. \( (g \circ f)(3) \) Now, we need to evaluate \( (g \circ f)(3) \): \[ (g \circ f)(3) = \sqrt{3} + 6 \] So, the answer is: \[ (g \circ f)(3) = \sqrt{3} + 6 \] ### Summary of Results: - a. \( (f \circ g)(x) = \sqrt{x + 6} \) - b. \( (g \circ f)(x) = \sqrt{x} + 6 \) - c. \( (f \circ g)(3) = 3 \) - d. \( (g \circ f)(3) = \sqrt{3} + 6 \)