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