Question

I'm really struggling with this form if you could leave steps that'd be amazing!

Answer 1

`2x + 5 + h`

Explanation:

You are told f(x) is defined this way.

`f(x) = x^2 + 5x - 3`

To find f(x + h), replace x with x + h in the definition of the function.

`f(x + h) = (x + h)^2 + 5(x + h) - 3` Equation 1

Now you need to insert this entire expression in the fraction you are given and simplify it.

Here is the expression you were given:

`(f(x + h) - f(x))/(h)`

Now we replace f(x + h) with the expression above (Eq. 1). We replace f(x) with the original quadratic expression.

`((x + h)^2 + 5(x + h) - 3 - (x^2 + 5x - 3))/(h)`

Now we expand the square of the binomial and distribute the negative throughout the parentheses.

`(x^2 + 2hx + h^2 + 5x + 5h - 3 - x^2 - 5x + 3))/(h)`

2x + h + 5

Combine like terms in the numerator.

`(2hx + h^2 + 5h))/(h)`

`(h(2x + h + 5))/(h)`

`2x + 5 + h`