In mathematics, functions are a fundamental concept that is used to describe the relationship between two sets of values. They are often represented as graphs, which can provide a visual representation of the relationship between the two sets. One interesting property of functions is whether they are even or odd. In this article, we’ll explore how to tell if a function is even or odd.
What is an Even Function?
An even function is a type of function that has a special property. If you were to graph an even function, you would notice that the graph is symmetric about the y-axis. This means that if you were to reflect the graph over the y-axis, it would look the same. Mathematically, we can define an even function as:
f(x) = f(-x)
In other words, if we plug in the opposite of any value of x into the function and get the same result, then the function is even. Let’s take a look at an example:
f(x) = x^2
If we plug in -x for x, we get:
f(-x) = (-x)^2 = x^2
As you can see, the result is the same as if we had plugged in x. Therefore, the function f(x) = x^2 is even.
What is an Odd Function?
An odd function is another type of function that has a special property. If you were to graph an odd function, you would notice that the graph is symmetric about the origin. This means that if you were to reflect the graph over the origin, it would look the same. Mathematically, we can define an odd function as:
f(x) = -f(-x)
In other words, if we plug in the opposite of any value of x into the function and get the negative of the result, then the function is odd. Let’s take a look at an example:
f(x) = x^3
If we plug in -x for x, we get:
f(-x) = (-x)^3 = -x^3
As you can see, the result is the negative of what we would get if we had plugged in x. Therefore, the function f(x) = x^3 is odd.
How to Tell If a Function is Even or Odd
Now that we know what even and odd functions are, let’s take a look at how we can tell if a function is even or odd. There are two methods we can use: graphical and algebraic.
Graphical Method
The graphical method involves graphing the function and determining if it is symmetric about the y-axis or the origin. If it is symmetric about the y-axis, it is even. If it is symmetric about the origin, it is odd. Let’s take a look at some examples:
Example 1
f(x) = x^4 – 2x^2
If we graph this function, we can see that it is symmetric about the y-axis. Therefore, it is even.
Example 2
f(x) = x^5 – x
If we graph this function, we can see that it is symmetric about the origin. Therefore, it is odd.
Algebraic Method
The algebraic method involves using the definition of even and odd functions to determine if a function is even or odd. Let’s take a look at some examples:
Example 1
f(x) = 3x^2 – 5
To determine if this function is even or odd, we need to plug in -x for x and see if we get the same result as if we had plugged in x. If we do, it is even. If we get the
