You've probably explored even and odd numbers in the past, but what about even and odd functions? Yes, they exist. No, determining whether a function is even or odd isn't as easy as checking whether the first term is an even or odd number. In this article, we'll see what makes a function even or odd and what that means when it's graphed on the Cartesian plane. Let's get started!
A function is considered even if, for each x in the domain . For instance:
We can test whether this is an even function by subbing -x for x and seeing if we get the same equation.
This is indeed an even function. When graphed, even functions have reflective symmetry across the y-axis as illustrated below:
Remember that a function is only even if every potential value of x satisfies the criteria. We cannot find a single value for x where and call the function even.
A function is considered odd if, for each x in the domain of f, . Again, we can test it algebraically:
is indeed an odd function. When graphed, odd functions have 180° rotational symmetry around the origin as shown:
The vast majority of functions won't satisfy the criteria for even or odd functions and are therefore neither. Never assume that a function must be even or odd.
a. Define a function as . Is this function even, odd, or neither?
We need to find f(-x) to determine its symmetry. If , the function is even. If , the function is odd. Otherwise, the function is neither even nor odd. Let's substitute -x for x:
We can simplify this using the Power of a Product Property:
Comparing this with , we find
Since , the function is odd.
b. Define a function . Is this function even, odd, or neither?
We're working with logarithms this time, but that doesn't change what we have to do. First, we need to determine what is by subbing -x in for x:
Since , this is an even function.
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