for the function f(x)= - 6x² + 2x - 1, evalua...
for the function f(x)= - 6x² + 2x - 1, evaluate and fully simplify each of the following. f(x + h)= (f(x + h)-f(x))/h =
Answer
# 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:
$f(x + h)=-6x^{2}-12xh-6h^{2}+2x + 2h-1$
$\frac{f(x + h)-f(x)}{h}=-12x-6h + 2$