Even Function: Definition & Examples

Instructor: Jasmine Cetrone

Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.

Even functions have properties of symmetry that we can take advantage of when graphing or analyzing these functions. Learn what makes a function even, how to recognize even functions, and see some examples!

What Makes Even Functions So Special?

Jesse is in grade school and wants to draw a heart for Valentine's Day, but he wants it to be a perfect one. What does he do? He folds his paper in half, draws one side of the heart in pencil, folds the paper, and rubs his finger over the paper to transfer some of the pencil markings onto the other side. When he unfolds it, he has a perfectly symmetric heart! Even functions share the same property as Jesse's drawing of a heart - that along some line (the y-axis, in this case) the left and right hand side look identical.

The heart Jesse drew has left-right symmetry
Symmetry about y-axis of a heart

A function defines a relationship between two variables (often x and y) where one variable depends on another. When we say y is a function of x, we are saying that when we input one value for x into the function, the function will output one (and only one) value for y. For example when we scan a barcode at the grocery store, we get a price for that item. When a box of cereal is scanned, the input is the barcode. The single price is then outputted. This would mean that price is a function of the barcode.

It's important to note that in order for this to be a function, we can have only one output. It wouldn't make sense to scan a box of cereal and get two prices, would it? No! Only one price should be listed for each item. This ensures that not only do we have a relationship, but that the relationship is a function.

Defining Even Functions

There are two ways to describe even functions. One is graphically, the other is algebraically.

If we look at the graph of an even function, we will notice that the graph looks identical to the left and to the right of the y-axis. In other words, the y-axis acts like a mirror to the function. Graphs of functions with y-axis symmetry are shown below.

Graphs of functions that display symmetry about the vertical axis are even functions
graphs that have left-right symmetry are even

Say that you are walking along the x-axis. You start at the origin and move 5 feet to the right and stop. You then look directly up (or down) to the graph of the function and take note of its height (the y value of the function when x=5). You then go back to the origin and move 5 feet to the left and stop. You are now at x=-5. When you look at the graph again, the function will have the same y-value as when x was at positive five! This helps us understand the algebraic definition of an even function.

An even function written algebraically would look like this:

f(x)=f(-x)

Meaning that the y-value of the function is the same at positive x and negative x.

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