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)=36x^2 - 72x + 33 (simplify your answer.) c. (f ∘ g)( - 2)= (simplify your answer.)

Explanation:

Step1: Recall the definition of composition

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

Step2: Simplify the expression for $f(g(x))$

\[

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

\]

Step3: Recall the definition for $(g\circ f)(x)$

$(g\circ f)(x)=g(f(x))$. Since $f(x)=3x - 3$, then $g(f(x))=4(3x - 3)^{2}-3$.

Step4: Expand and simplify $(3x - 3)^{2}$

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

Step5: Substitute and simplify $g(f(x))$

\[

$$\begin{align*} g(f(x))&=4(9x^{2}-18x + 9)-3\\ &=36x^{2}-72x+36 - 3\\ &=36x^{2}-72x + 33 \end{align*}$$

\]

Step6: Find $(f\circ g)(-2)$

Substitute $x = - 2$ into $(f\circ g)(x)=12x^{2}-12$. So $(f\circ g)(-2)=12\times(-2)^{2}-12$.

Step7: Calculate the value of $(f\circ g)(-2)$

\[

$$\begin{align*} (f\circ g)(-2)&=12\times4-12\\ &=48-12\\ &=36 \end{align*}$$

\]

Answer:

a. $(f\circ g)(x)=12x^{2}-12$
b. $(g\circ f)(x)=36x^{2}-72x + 33$
c. $(f\circ g)(-2)=36$
d. First, find $(g\circ f)(x)=36x^{2}-72x + 33$. Then substitute $x=-2$:
\[

$$\begin{align*} (g\circ f)(-2)&=36\times(-2)^{2}-72\times(-2)+33\\ &=36\times4 + 144+33\\ &=144+144+33\\ &=321 \end{align*}$$

\]
So $(g\circ f)(-2)=321$