[SOUND] Let's learn about odd and even functions.

[SOUND] For example, the entire graph of three different functions is shown below.

Let's determine whether each function is odd, even, or neither.

[SOUND] A function is called even if f(-x)=f(x) for all f's in the domain.

And the graph of and even function is symmetric with respect to the y-axis.

For example, f(x)=x^2 is a famous example of an even function.

And if you notice here, that whenever x, f(x) is on the graph, then -x, f(-x) is

also on the graph where f(-x)=f(x). That is, plugging in opposite x values

yields the same y value. And therefore, this part of the graph is

a mirror image of this part of the graph. That is, the graph is symmetric with

respect to the y-axis. And a function is called odd if

f(-x)=-f(x) for all x in the domain. And the graph of an odd function is

symmetric with respect to the origin. For example, f(x)=x^3 is a famous example

of an odd function. And notice, whenever x, f(x) is on the

graph, and so is negative -x, f(-x).

But moreover, f(-x)=-f(x). In other words, when the x values are

opposites, so are the y values. Which means, that this part of the graph

here is a mere image of a part of the graph here.

In other words, its graph is symmetric with respect to the origin.

And so, if we have the graph of the function, we can look at whether or not

its graph is symmetric with respect to the y-axis or the origin or neither, to

determine whether it's even, odd or neither.

But if we don't know the graph and only are given the equation, then we can see

if plugging in opposite x values yields the same y value, or opposite y values,

or neither. So, in this first example here, we're

given the graph of three different functions.

So, let's look at this first function here.

We see that this part of the graph is a mirror image of this part of the graph,

that is its graph is symmetric with respect to the y-axis.