(a) find ( f(x + h) ) and simplify. (b) find ( \frac{f(x + h) - f(x)}{h} ) and simplify. part 1 of 2 (a) ( f(x + h) = x^{2}+2 x h+h^{2}-3 x - 3 h )

Question

Accepted Answer

# Explanation:
## Step1: Substitute $x + h$ into $f(x)$
Assume $f(x)=x^{2}-3x$. Then $f(x + h)=(x + h)^{2}-3(x + h)$
## Step2: Expand the expressions
$$
\begin{align*}
(x + h)^{2}-3(x + h)&=x^{2}+2xh+h^{2}-3x-3h
\end{align*}
$$
## Step3: Calculate $\frac{f(x + h)-f(x)}{h}$
Substitute $f(x + h)=x^{2}+2xh+h^{2}-3x-3h$ and $f(x)=x^{2}-3x$ into $\frac{f(x + h)-f(x)}{h}$:
$$
\begin{align*}
\frac{(x^{2}+2xh+h^{2}-3x-3h)-(x^{2}-3x)}{h}&=\frac{x^{2}+2xh+h^{2}-3x-3h - x^{2}+3x}{h}\
&=\frac{2xh+h^{2}-3h}{h}\
&=\frac{h(2x + h-3)}{h}\
&=2x+h - 3
\end{align*}
$$

# Answer:
(a) $f(x + h)=x^{2}+2xh+h^{2}-3x-3h$
(b) $\frac{f(x + h)-f(x)}{h}=2x+h - 3$

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QUESTION IMAGE

Question

Explanation:

Step1: Substitute \(x + h\) into \(f(x)\)

Step2: Expand the expressions

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

Answer: