### Points, graphs, and function notation

Ah, function notation, that old SAT math bugaboo. I often find that otherwise strong students still struggle with questions that combine graphs and function notation. I'm here to tell you today that, if you're willing to put in a few minutes of focused practice here, you'll need not fear these questions any longer.  Let's just get right into it with an example.

1. The figure above shows the graph of f(x) from x = –6 to x = 6. If f(3) = p, what is f(p)?

(A) –4
(B) –2
(C) 2
(D) 3
(E) 6

To solve a question like this, you must remember a simple but important fact: "f(a) = b" is just shorthand for saying "the function f contains the point (a, b)." So when this question tells you that f(3) = p, it's telling you that the function f contains the point (3, p). All you need to do is go to the graph and see where it is when x = 3.

When x = 3, the graph of the function is at (3, 6). So f(3) = 6, and therefore p = 6.

And now you're almost done. To finish up, just go to the graph one more time to find f(p), which you now know is really just f(6).

Becoming deft with questions like this just requires practice. So...you should do some practice. Let's use the same graph, and I'll ask you a few more questions, increasing the complexity as I go. Hover your mouse over the questions (or tap them, if you're on a tablet or phone) and the answer will appear like magic.

• If f(4) = a, what is f(a)?
• If f(–2) = b, what is f(b)?
• If f(–1) = e, what is f(e) – 6?
• If f(1) = g, what is f(–g)?
• If f(5) = c, what is f(c + 1)?
• If f(2) = d, what is 2f(2d)?
• If f(6) = m and f(2) = n, what is 3f(mn)?
• If f(–3) = r, what is f(f(r))?
These are super fun, right?