Question

If f(x)=x^2+3x+4 which of the number choices represents (f(x+h)-f(x))/h?

This a precalculus equation from a caching activity. I would definitely appreciate an explanation as well as an answer to the question.

Answer 1

In case you're not already aware, the expression
`\frac{f(x+h)-f(x)}h` is called the "difference quotient" and represents the average rate of change of a function
`f` over an interval
`[x,x+h]`.

For the function
`f(x)=x^2+3x+4`, by substituting
`x+h` we get

`f(x+h)=(x+h)^2+3(x+h)+4=(x^2+2xh+h^2)+(3x+3h)+4=x^2+3x+4+(2x+3)h+h^2`

Then the difference quotient is

`\frac{f(x+h)-f(x)}h=\frac{(x^2+3x+4+(2x+3)h+h^2)-(x^2+3x+4)}h`

`=\frac{(2x+3)h+h^2}h=2x+3+h`

where the last equality holds as long as
`h\\eq0`.