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

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x): [ \begin{align*} f(x + h)&=(x + h)^2-3(x + h)+5\ &=x^{2}+2xh+h^{2}-3x-3h + 5 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-3x-3h + 5)-(x^{2}-3x + 5)\ &=x^{2}+2xh+h^{2}-3x-3h + 5-x^{2}+3x - 5\ &=2xh+h^{2}-3h \end{align*} ]

Step3: Find the difference - quotient

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

Answer:

$2x+h - 3$