# ACT Math: Even and Odd Functions

The ACT Math section always contains a few trigonometry questions that go beyond the basic SOH-CAH-TOA problems that you learned in geometry.  Sometimes these problems have to do with the properties of functions, and one of the concepts you might be tested on is whether a function is even or odd.

So what does it mean for a function to be even or odd?

An even function is one for which f(x) = f(-x) for all values of x. That’s a pretty “mathy” definition, so what does it mean? Another way of saying the same thing is “opposite x‘s give you the same y.” An example of an even function that you’re probably familiar with is f(x) = x ² + 3. You can pick any number for x and f(x) = f(-x). For instance, let’s try x = 2.

f(x) = x² + 3
f(2) = 2² + 3 = 4 + 3 = 7
f(-2) = (-2)² + 3 = 4 + 3 = 7

You can also see this if you look at the graph of the function. One thing you may also notice is that an even function is symmetrical with the y-axis. In trigonometry, a graph that you’ll encounter that is even is f(x) = cos(x).

An odd function is on for which f(-x) = –f(x) for all values of x. In other words, opposite x‘s give you opposite y‘s. An example of a function that is odd is f(x) = x³ – 3x. Let’s try x = 2 again and see what happens.

f(x) = x³ – 3x
f(2) = 2³ – 3(2) = 8 – 6 = 2
f(-2) = (-2)³ – 3(-2) = -8 + 6 = -2

Just as with the even function above, you can see this in the graph. You might also notice (although it’s probably less intuitive than the y-axis symmetry of the even function) that an odd function is symmetrical with the origin. In trigonometry, a graph that you’ll encounter that is odd is f(x) = sin(x).

So let’s have you try a couple of problems similar to those that might appear on the ACT (probably near the end of the math section). Solutions appear at the bottom of the page.

1.  The graph of f(x) = 2cos(3x) is shown below. Which of the following statements about the function is true? A.  The function is even (that is f(x) = f(-x) for all values of x.)
B.  The function is odd (that is f(-x) = –f(x) for all values of x.)
C.  The function is neither even nor odd
D.  The function is undefined at x = 0
E.  There is no value of x for which f(x) = 1.

2.  The graph of f(x) = sin(πx) is shown below. Which of the following statements about the function is true? F. The function is even (that is f(x) = f(-x) for all values of x.)
G. The function is odd (that is f(-x) = –f(x) for all values of x.)
H. The function is neither even nor odd
J. The function is undefined at x = π
K.  The range of the function is (-∞, ∞ )

If you have questions about these problems or anything else to do with the ACT, drop us an email at info@cardinalec.com

Solutions:
1.  A
2.  G