EvEn anD oDD FuNcTiONs

Let's talk about functions. To start off, a function which can be

represented by f(x) is said to be a function because for every input x (DOMAIN) that we put in a function there is always just one single output which is y (RANGE). So, the domain of a function represents all the possible inputs for the x value and the range represents the outputs of the function, which is the value of y or f(x).



There are two types of functions even and odd. An even function is a function that is symmetrical about the y-axis meaning that one side of the function reflects across the y-axis. This type of function is represented by f(-x)=f(x). This means that if you plug -x for every x in the function you end up having the same function that you started with.

For example:

f(x)= -3x^2 + 4

f(-x)= -3(-x)^2 + 4

= -3(x^2) + 4

= -3x^2 + 4 -------you end up with the same

function you started with :)

The graph of that even function is the one below

- We can see that the right side of the function on quadrant 1 reflects on the other, quadrant 2. A characteristic of even functions is that a point (x,y) is on the graph only if the point

(-x, y) it's on it too. In this case we can see that the points (1,1) and (-1,1) are on the graph.

An odd function is a function that is symmetrical about the origin meaning that the function reflects across the y-axis and across the x-axis at the third quadrant. This type of function is represented by f(-x)= -1f(x). This means that if you plug -x for every x in the function the function will end up looking with all its signs opposite to what they were in the original function due to the -1.

For example:

f(x)= 2x^3 - 4x

f(-x)= 2(-x)^3 - 4(-x)

= 2(-x^3) + 4x

= -2x^3 + 4x ------------------- we end up having the opposite signs. It's

like putting a -1 in front ( -1f(x)) and

multiply everything by it.

The graph of the function above is this one

- For odd functions it's kind of hard to tell the reflection across the y- axis and at the third quadrant but it's there. In this kind of functions a point (x, y) is on the graph only if the point (-x, -y), which corresponds to the third quadrant, is there too. In this case the point (2, 8) and (-2, -8) are on the graph.

The odd function below shows the reflections across the y-axis and at the third quadrant.

If you would like to prove these reflections you can do it by tracing the graph of the function on butter paper.

You fold the paper in half at the y-axis of the graph and you can see that the function reflects across the y-axis having a mirror image and the reflection across the x-axis at the third quadrant.

This is f(x)=sinx

NOTE: A function can also be neither even nor odd. This happens when you plug in -X for every X in the function and at the end you end up neither with the same function you started with nor a function with all the signs opposite to the original.