Question

For the function f(x) = x2 - 4. construct and simplify the difference quotient f(x+h)-f(x) h

Answer 1

$\begin{gathered} f(x)=x^2-4 \\ \frac{f(x+h)-f(x)}{h}\text{ = }\frac{(x+h)^2-4-(x^2-4)}{h} \\ \end{gathered}$$=\frac{(x+h)(x+h)-4-x^2+4}{h}\text{ = }\frac{x^2+hx+hx+h^2-x^2-4+4}{h}$$=\frac{x^2+2hx+h^2-x^2}{h}$$\begin{gathered} =\frac{x^2-x^2+2hx+h^2}{h} \\ \\ =\frac{2hx+h^2}{h} \\ \\ =\frac{h(2x+h)}{h} \\ \\ \frac{f(x+h)-f(x)}{h}\text{ }=2x+h \end{gathered}$