Question

for f(x)=√x and g(x)=x + 1, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(3); d. (g ∘ f)(3)
a. (f ∘ g)(x)=√(x + 1) (simplify your answer.)
b. (g ∘ f)(x)=√x + 1 (simplify your answer.)
c. (f ∘ g)(3)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

$(f\circ g)(x)=f(g(x))$. Since $g(x)=x + 1$ and $f(x)=\sqrt{x}$, then $(f\circ g)(x)=f(x + 1)=\sqrt{x + 1}$.

Step2: Recall other - composition formula

$(g\circ f)(x)=g(f(x))$. Since $f(x)=\sqrt{x}$ and $g(x)=x + 1$, then $(g\circ f)(x)=g(\sqrt{x})=\sqrt{x}+1$.

Step3: Calculate $(f\circ g)(3)$

Substitute $x = 3$ into $(f\circ g)(x)$. We know $(f\circ g)(x)=\sqrt{x + 1}$, so $(f\circ g)(3)=\sqrt{3+1}=\sqrt{4}=2$.

Step4: Calculate $(g\circ f)(3)$

Substitute $x = 3$ into $(g\circ f)(x)$. We know $(g\circ f)(x)=\sqrt{x}+1$, so $(g\circ f)(3)=\sqrt{3}+1$.

Answer:

a. $(f\circ g)(x)=\sqrt{x + 1}$
b. $(g\circ f)(x)=\sqrt{x}+1$
c. $(f\circ g)(3)=2$
d. $(g\circ f)(3)=\sqrt{3}+1$