for the function f(x)=3x² - x - 3, evaluate and fully simplify each of the following. f(x + h)= f(x +…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=3(x + h)^2-(x + h)-3\ &=3(x^{2}+2xh+h^{2})-x - h-3\ &=3x^{2}+6xh+3h^{2}-x - h-3 \end{align*} ]

Step2: Find $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=(3x^{2}+6xh+3h^{2}-x - h-3)-(3x^{2}-x - 3)\ &=3x^{2}+6xh+3h^{2}-x - h-3 - 3x^{2}+x + 3\ &=6xh+3h^{2}-h \end{align*} ]

Step3: Find $\frac{f(x + h)-f(x)}{h}$

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{6xh+3h^{2}-h}{h}\ &=\frac{h(6x + 3h-1)}{h}\ &=6x+3h - 1 \end{align*} ]

Answer:

$f(x + h)=3x^{2}+6xh+3h^{2}-x - h-3$ $\frac{f(x + h)-f(x)}{h}=6x+3h - 1$