Hi everyone.
Today we’re going to talk about how to determine whether a function is an even function, an
odd function or neither an even nor odd function.
To complete this problem, we will substitute negative x for x, simplify and then see whether
our results is equal to our original function and even, equal to our original function times
negative 1 and odd or neither.
Let’s take a look.
In this particular problem, we have been given three separate functions and asked to determine
whether each one is an even function, an odd function or neither.
To make this determination, in each case, we will plug in negative x in place of x to
our functions, simplify and then the result will tell us whether the function is even,
odd or neither even or odd.
Let’s go ahead first and plug in negative x to our first function.
We will get negative x to the fifth plus negative x and again, we just want to simplify as much
as we can here.
Negative x to the fifth will give us negative x to the fifth like this minus x.
At first glance, this might not look the same as our original function f of x but if we
factor out a negative sign, we will get negative x to the fifth plus x and we can see that
our functions are the same but for the negative sign out in front.
We have in our original function x to the fifth plus x and then our simplified function
here x to the fifth plus x.
The only difference here is this negative sign.
As you can see, since we know that f of x is equal to x to the fifth plus x, we could
just as easily substitute f of x for x to the fifth plus x and you can see that we get
negative f of x.
That’s exactly what we were looking for to determine that this is an odd function
so we know here that we have an odd function.
We will go ahead and do the same thing with our second function.
As always, we will plug in negative x in place of x and we will get 1 minus negative x to
the fourth.
When we simplify this, we will get 1 minus.
Negative x to the fourth will give us a positive sign.
That negative sign inside the parenthesis will go away and we will just be left with
x to the fourth.
As you can see really quickly, we’re right back where we started at f of x.
So this just becomes f of x and as you know, when we get back to our original function
f of x, we know that that’s an even function.
We have defined these down below by the way.
We can see that if we get back to f of x, we have an even function.
If we get to negative f of x as we did in the previous problem, we have an odd function
and then when neither of those things is true, we get neither even nor odd.
Let’s go ahead and look at our third function.
Again, as always, we will plug in negative x for x.
That’s always how we make our determination of even, odd or neither.
So we will see that we get two times negative x minus negative x squared.
We will simplify this as much as we can and we will get negative 2x minus this negative
x squared.
That negative sign inside the parentheses will go away and we’ll be left with x squared.
As you can see, this isn’t equal to f of x.
They’re exactly the same except for this negative sign out in front of the 2x here
but we didn’t get f of x exactly so we can’t call it even.
Even if we try to factor out a negative sign and we get negative times 2x plus x squared,
we still don’t have our original function here because our original function is 2x minus
x squared and we have 2x plus x squared.
So we can’t call this odd either, which means this is going to be neither and that’s
it.
That’s all there is to it when it comes to determining whether or not your function
is even, odd or neither even nor odd.
So I hope you found that video helpful.
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