Question

For the function f(x)= x^2+8x, how do you find and simplify f(x+h)-f(x)?

Answer 1

`h^2+2xh-8x`

Explanation:

`f(x)=x^2+8x` for variable unknown `x`

If the variable unknown is `x+h` then the function `f` will become:
`f(x+h)=(x+h)^2+8(x+h)`

Thus, `f(x+h)-f(x)` is:

`[(x+h)^2+8(x+h)]-[x^2+8x]`
`[(x+h)(x+h)+8x+8h]-x^2-8x`
`[x^2+2xh+h^2]-x^2-8x`

Cancel out any corresponding terms:
`\cancel{x^2}+2xh+h^2\cancel{-x^2]-8x\rightarrow\color{red}{h^2+2xh-8x}`