Hi, Kristin.

f(x) = x^{2} - 1 is a function, which just tells us what to do with x.

Whatever is inside the parentheses gets substituted for x in the equation. For example:

f(3) = 3^{2} - 1 so f(3) = 8

Then f(x-3) means that we substitute x-3 in for x, and simplify:

f(x-3) = (x - 3)^{2} -1

We can multiply (x-3)(x-3) and get:

f(x-3) = (x^{2} -6x + 9) - 1 which simplifies to:

**f(x-3) = x**^{2} - 6x + 8

The second expression has two parts, and I will do the f(4+h) first:

f(4+h) = (4 + h)^{2} -1

Mutiplying and simplifying gives us: f(4+h) = h^{2} + 8h + 15

f(4) = 4^{2} - 1 --> f(4) = 15

So combining them:

f(4+h) - f(4) = h^{2} + 8h + 15 -15

f(4+h) - f(4) = h^{2} + 8h which factors to:

**f(4+h) - f(4) = h(h + 8)**

Hope this helps!

Kathye P.

Oops, you're right! Thanks. I will correct it.

10/02/12