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)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

The composition \((f\circ g)(x)=f(g(x))\). Given \(f(x) = 3x - 3\) and \(g(x)=4x^{2}-3\), we substitute \(g(x)\) into \(f(x)\).

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

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

Step3: Expand and simplify

\[

$$\begin{align*} f(g(x))&=3\times4x^{2}-3\times3 - 3\\ &=12x^{2}-9 - 3\\ &=12x^{2}-12 \end{align*}$$

\]

Answer:

\(12x^{2}-12\)