find the difference quotient of f; that is, find (f(x + h)-f(x))/h, h≠0, for the following function. f(x)=x²…

Explanation:

Step1: Find f(x + h)

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

Step2: Calculate f(x + h)-f(x)

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

Step3: Find the difference - quotient

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

Answer:

(2x+h - 2)