Question

How do you evaluate the function `f (x) = 3x^2 + 3x -2` for `f (a + h)`?

Answer 1

Plug in `(a+h)` for `x`.

Explanation:

If the question asked for `f(2)`, all you would do is put `2` into every spot where there's an `x`. For `(a+h)`, you use the same logic, and plug `(a+h)` into every spot where there's an `x`.

`color(blue)(f(a+h)=3(a+h)^2+3(a+h)-2)`

This quickly turns into a question of algebra. Consider the first term, `3(a+h)^2`. The order of operations states that we must square `(a+h)` before we distribute the `3`.

So, in order to square `(a+h)`, we "FOIL" the following: `(a+h)(a+h)`.
We should receive `a^2+ah+ah+h^2`, which is equivalent to `a^2+2ah+h^2`.

Therefore, we now have `color(green)(f(a+h)=3(a^2+2ah+h^2)+3(a+h)-2)`
Now, we can distribute the `3` into both terms in parentheses.
We should get `f(a+h)=3a^2+6ah+3h^2+3a+3h-2`.

Now, all we have to do is combine like terms—but wait! There are none. Everything is as simplified as possible. Therefore, we have arrived at our answer.

`color(red)(f(a+h)=3a^2+6ah+3h^2+3a+3h-2)`