Question

for f(x)=x² + 9 and g(x)=x² - 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)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

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

Step2: Substitute $g(x)$ into $f(x)$

$f(g(x))=(x^{2}-3)^{2}+9$.

Step3: Expand $(x^{2}-3)^{2}$

Using $(a - b)^{2}=a^{2}-2ab + b^{2}$, where $a = x^{2}$ and $b = 3$, we have $(x^{2}-3)^{2}=(x^{2})^{2}-2(x^{2})(3)+3^{2}=x^{4}-6x^{2}+9$.

Step4: Simplify the expression

$f(g(x))=x^{4}-6x^{2}+9 + 9=x^{4}-6x^{2}+18$.

Answer:

$x^{4}-6x^{2}+18$