Question

for f(x)=x^2 + 9 and g(x)=x^2 - 3, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(4); d. (g ∘ f)(4) a. (f ∘ g)(x)=x^4 - 6x^2 + 18 (simplify your answer.) b. (g ∘ f)(x)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

$(g\circ f)(x)=g(f(x))$. Given $f(x)=x^{2}+9$ and $g(x)=x^{2}-3$, substitute $f(x)$ into $g(x)$.

Step2: Substitute and simplify

$g(f(x))=(x^{2}+9)^{2}-3$. Expand $(x^{2}+9)^{2}$ using $(a + b)^{2}=a^{2}+2ab + b^{2}$ where $a=x^{2}$ and $b = 9$. So $(x^{2}+9)^{2}=(x^{2})^{2}+2\times x^{2}\times9+9^{2}=x^{4}+18x^{2}+81$. Then $(x^{2}+9)^{2}-3=x^{4}+18x^{2}+81 - 3=x^{4}+18x^{2}+78$.

Answer:

$x^{4}+18x^{2}+78$