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ACT Prep: Help and Review44 chapters  435 lessons  26 flashcard sets
Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.
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 yaxis, in this case) the left and right hand side look identical.
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.
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 yaxis. In other words, the yaxis acts like a mirror to the function. Graphs of functions with yaxis symmetry are shown below.
Say that you are walking along the xaxis. 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 yvalue 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:
Meaning that the yvalue of the function is the same at positive x and negative x.
If you want to determine if you have an even functionâ€¦how do you do it? Pretty simple! Let's take advantage of the property f(x)=f(x). We'll look at the rule (or equation) for the function and replace x with x. If, after we simplify, we get back to our original function f(x), it means that f(x)=f(x) and we have an even function!
Example #1:


We notice that using algebra, f(x)=f(x) which indicates that f(x)=x^2 is an even function. Additionally, the graph displays yaxis symmetry, which is also consistent with an even function.
Example #2:


Carefully working with the properties of absolute value functions, we see that g(t)=g(t). This graph also shows symmetry about the yaxis. Both show that g(t)=t is an even function.
Example #3


In this case, we do not have the necessary condition on h(x) to be an even function. Also notice that the graph does not have symmetry about the yaxis, also indicating that it is not an even function.
Example #4


Now if you haven't dealt with trigonometry yet, that's ok! If you take a look at the graph above, you'll notice the symmetry about the vertical axis. This suggest that cos(x) is an even function, too! So it has the property cos(x)=cos(x).
Even functions can be determined algebraically by using the property f(x)=f(x) or graphically by noticing symmetry about the yaxis. By remembering the basic properties of even functions, they are easy for us to identify!
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ACT Prep: Help and Review44 chapters  435 lessons  26 flashcard sets