for f(x) = √x and g(x) = x + 3, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(6)…

for f(x) = √x and g(x) = x + 3, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(6); d. (g ∘ f)(6) a. (f ∘ g)(x) = (simplify your answer.) b. (g ∘ f)(x) = (simplify your answer.) c. (f ∘ g)(6) = (simplify your answer.) d. (g ∘ f)(6) = (simplify your answer.)

for f(x) = √x and g(x) = x + 3, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(6); d. (g ∘ f)(6) a. (f ∘ g)(x) = (simplify your answer.) b. (g ∘ f)(x) = (simplify your answer.) c. (f ∘ g)(6) = (simplify your answer.) d. (g ∘ f)(6) = (simplify your answer.)

Answer

Explanation:

Step1: Recall function - composition definition

The composition ((f\circ g)(x)=f(g(x))). Given (f(x)=\sqrt{x}) and (g(x)=x + 3), then ((f\circ g)(x)=f(g(x))=f(x + 3)=\sqrt{x + 3}).

Step2: Recall the other - direction of function - composition

The composition ((g\circ f)(x)=g(f(x))). Since (f(x)=\sqrt{x}) and (g(x)=x + 3), then ((g\circ f)(x)=g(\sqrt{x})=\sqrt{x}+3).

Step3: Evaluate ((f\circ g)(6))

Substitute (x = 6) into ((f\circ g)(x)). We know ((f\circ g)(x)=\sqrt{x + 3}), so ((f\circ g)(6)=\sqrt{6 + 3}=\sqrt{9}=3).

Step4: Evaluate ((g\circ f)(6))

Substitute (x = 6) into ((g\circ f)(x)). We know ((g\circ f)(x)=\sqrt{x}+3), so ((g\circ f)(6)=\sqrt{6}+3).

Answer:

a. (\sqrt{x + 3}) b. (\sqrt{x}+3) c. (3) d. (\sqrt{6}+3)