Even Odd or Neigher Function

As we have learned what is a function, we will know most of the time we will need to graph it out, so now we are going to start with the basic kind of feature of function.

Even Function and Odd Function

So you may have some questions such as what are even and odd functions. We will start with an example to explain. Let's say we have two functions

1. f(x)=$x^{2}$
2. p(x)=$x^{3}$

The first one is f of x equals x to the power of two, the second one is p of x equals x to the power of 3. So in this example, the first one will be an even function because the biggest exponent of x is two, and two is an even number. The second one will be an odd function because the biggest exponent of x is three, and three is an odd number. So it will be very easy to know what kind of function it is because you just need to look at what kind of number is the biggest exponent of x then you can tell if it is even or odd. There also could be neither function but we will talk about it later.

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Properties of even function

1. The sum of two even functions is even. The difference between the two even functions is even.
2. The product of two even functions is even.
3. The quotient of the division of two even functions is even. The composition of two even functions is even.
4. The composition of an even and odd function is even.

Properties of odd function

1. The sum of two odd functions is odd.
2. The difference between two odd functions is odd.
3. The product of two odd functions is even.
4. The quotient of the division of two odd functions is even.
5. The composition of two odd functions is odd.
6. The composition of an even function and an odd function is even.

The examples for neither function

When a function is not symmetric about the y-axis, and not symmetric about the origin, that function will be neither function for example tangent function in trigonometry. Even the function has no symmetry. It’s possible that a graph could be symmetric to the x-axis, but then it wouldn’t pass the Vertical Line Test and therefore wouldn’t be a function.