for the function f(x)= - 6x² + 2x - 1, 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)&=-6(x + h)^2+2(x + h)-1\ &=-6(x^{2}+2xh+h^{2})+2x + 2h-1\ &=-6x^{2}-12xh-6h^{2}+2x + 2h-1 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-6x^{2}-12xh-6h^{2}+2x + 2h-1)-(-6x^{2}+2x-1)\ &=-6x^{2}-12xh-6h^{2}+2x + 2h-1 + 6x^{2}-2x + 1\ &=-12xh-6h^{2}+2h \end{align*} ]

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

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-12xh-6h^{2}+2h}{h}\ &=\frac{h(-12x-6h + 2)}{h}\ &=-12x-6h + 2 \end{align*} ]

Answer:

$f(x + h)=-6x^{2}-12xh-6h^{2}+2x + 2h-1$ $\frac{f(x + h)-f(x)}{h}=-12x-6h + 2$