ACT Math Skill Review: Functions
A function is a rule or formula that tells how to relate elements in one set (the domain) with the elements in another set (the range). You are likely already familiar with many different types of function notations. Here are a few examples:
In each case, a specific value of x will give you a specific value of f(x) or g(x). Let’s take a closer look:
When you see a question like this, all you need to do is substitute the new x value into the function. In this case, substitute 2 for x in the function.
f(2)= (2)^{2 }+ 2(2)  1 = 4 + 4  1 = 7
You are likely to encounter more complex functions on the SAT as well. Here’s a quick look at a few of them:
In the above example, the function is split into two halves: the half that comes before x = 1 and the half that goes from x = 1 to infinity. Which half of the function you use depends on the value of x. If we evaluate f(0), we must use the first function, since 0 < 1. Then, f(0) = 2(0)^{2 } 1 = 1. If we evaluate f(1), we must use the second half of the function, since x ≥ 1. Therefore, f(1) = 1 + 4 = 5.
For this example, just plug in x + h everywhere you find an x. Thus:
f(x) = 3x^{2 }+ 2x
f(x+h) = 3(x+h)^{2 }+ 2(x+h)
= 3(x^{2 }+ 2xh + h^{2}) + 2x + 2h
= 3x^{2 }+ 6xh + h^{2 }+ 2x + 2h
When it comes to working with functions, if you’re feeling a little confused, just remember this basic tenet: Follow the rules. Pretty simple, right? Unfortunately, the ACT likes to mix things up and deviate from the standard f(x) or g(x) format for functions. They’ll get creative and show wacky symbols, like the following:
Don’t let the weird squiggle thing throw you off. This is just a function problem in disguise. Remember, just follow the rules. Everything to the left of the squiggle gets plugged in for x and everything to the right of the squiggle (2 + x) gets plugged in for y. They try to confuse you by reusing x. This is what you’re left with:
Examples
Answers and Explanations

The correct answer is C. Replace x with 4 to get: (4)^{2 } 3(4) + 12 = 16 + 12 + 12 = 40.

The correct answer is J. The square symbol is just a standin for a more typical function format. This question could just as easily been rewritten as f(x,y) = 15⁄x + 2x + 3y. Just replace x with 5 and y with 10 to get: 15⁄5 + 2(5) + 3(10) = 3 + 10 + 30 = 43.