Question

for f(x)=3x - 3 and g(x)=4x^2 - 3, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(-2); d. (g ∘ f)(-2) a. (f ∘ g)(x)=12x^2 - 12 (simplify your answer.) b. (g ∘ f)(x)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

$(g\circ f)(x)=g(f(x))$. Given $f(x) = 3x - 3$ and $g(x)=4x^{2}-3$.

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

Replace $x$ in $g(x)$ with $3x - 3$. So $g(f(x))=4(3x - 3)^{2}-3$.

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

Using $(a - b)^{2}=a^{2}-2ab + b^{2}$, where $a = 3x$ and $b = 3$, we have $(3x - 3)^{2}=(3x)^{2}-2\times3x\times3 + 3^{2}=9x^{2}-18x + 9$.

Step4: Multiply by 4 and subtract 3

$4(3x - 3)^{2}-3=4(9x^{2}-18x + 9)-3=36x^{2}-72x+36 - 3$.

Step5: Simplify the expression

$36x^{2}-72x+36 - 3=36x^{2}-72x + 33$.

Answer:

$36x^{2}-72x + 33$