Question

sec 3.4 combination and composition (1 point) let f(x)=3x^3 - 6 and g(x)=2 - 4x. (f ∘ g)(x)= help (formulas) preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining. email instructor

Explanation:

Step1: Recall composition formula

$(f\circ g)(x)=f(g(x))$.

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

Since $f(x) = 3x^{3}-6$ and $g(x)=2 - 4x$, we substitute $x$ in $f(x)$ with $2 - 4x$. So $f(g(x))=3(2 - 4x)^{3}-6$.

Step3: Expand $(2 - 4x)^{3}$

Using the formula $(a - b)^{3}=a^{3}-3a^{2}b + 3ab^{2}-b^{3}$ where $a = 2$ and $b = 4x$, we have $(2-4x)^{3}=2^{3}-3\times2^{2}\times(4x)+3\times2\times(4x)^{2}-(4x)^{3}=8-48x + 96x^{2}-64x^{3}$.

Step4: Multiply by 3 and subtract 6

$3(2 - 4x)^{3}-6=3(8-48x + 96x^{2}-64x^{3})-6=24-144x + 288x^{2}-192x^{3}-6=-192x^{3}+288x^{2}-144x + 18$.

Answer:

$-192x^{3}+288x^{2}-144x + 18$