We're going to talk about how to do it the easy way and also how to do tithe way. Your teacher wants you to do it so let's begin the first thing you need. How to tell if its even function is even if F of negative. X is equal to f of X So if you replace X with negative x and there's no change. The new function that you get looks exactly like the original function. Then it's even now what about if it's odd it's odd if Off negative x is equal to negative f. So that is if you replace negative which acts everything in the function every term has to change sign. If one term changes sign and the rest do not it’s not going to be odd. Now, what about the last category when is a function neither even nor odd. It's going to be neither if you plug in negative x and if you do not get negative f. So what does that mean well let's say if you replace X with negative x and some signs change while others?

Then it’s going to be neither also if it doesn’t equal f of
x2 it's also neither. It can't equal negative f of X or off X so basically as
long as is not even or oddity’s neither. Let's start with our first example
let's say that f of X is equal to X to the fourth plus 3x squared. The function
even odd or is it neither well here's. How you do it easily look at the
exponent is 4 even or odd 4 is an even number. Now what about 2 is it even or
odd 2 is, even so, all the exponents. Even then the function is going to be
even but now let’s prove it. Let's show our work let’s replace X with negative
x so this is going to be negative x raised to the fourth power plus 3 times.
Negative squared now what is negative x raised tithe fourth power. Negative x
times itself 4 times negative. Squared is** How To Tell If A Function Is Even Or Odd** negative x
times itself two times two negative is a positive. Three negatives will give
you a negative result but four. Negatives will give you positive results
anytime. You have an even number of negative signs. It's going to produce a
positive sign so this is going to be positive X to the fourth plus. Three x
squared to the negative science will produce a positive result. Now notice that
the function that we have is the same as the original function. Therefore this
is equal to f of so. We can make the statement that F of negative x is equal to
f of X which is the definition of an even function. Now let's try another
example let's say that f of X is equal to X to the fifth. Power plus 2x to the
third power is it even odd or neither now don't worry about the coefficient.

This is unimportant even though two is an even number that's
not going to helps determine. If it's even or odd look at the exponent five is
it even or odd five is an odd number. Three is also an odd number since all of
the exponents are odd the function is going to be odd. Let’s replace X with
negative x. Now, whenever you have an odd number of negative signs the result
will be negative. For example, negative X to the third power negative times
negative x is positive x squared times. Another negative x that's negative
cubed so this is going to be negative. To the fifth power minus two x cubed and
all of that is equal to F of negative x. Now, what should we do in our next
step to prove that this function is odd? What you want to do is you want to
factor **How To Tell If A Function
Is Even Or Odd** out a negative one. If you take out negative one
negative x to the fifth divided by negative one is positive x statists. All the
signs will change negative two Cube will become positive two x cube. Now notice
that this portion inside the brackets X to the fifth plus. Two X cubes equal to
the original function so at that point. What you want to do is replace it with
the original function. Off X we can say that F of negative x is equal to
negative f of Which is the definition of an odd function. How you can prove it
now what about this one.

Let's say that off X is equal to x squared plus 6 is it even or odd well. You know x squared is an even component. It has an even exponent what about 6 well 6 is the same as 6 X to the 0. Anything raised to the 0power is 1 so X to the 0 is 1 which 6times X to 0 is 6 times 1 at 6 and 0 is an even exponent. So the whole thing is going to be even so let's go ahead and prove it. Now let's replace X with negative x so this is going to be negative x squared plus 6 negative times. Negative x is positive x squared so we have x squared plus 6 notice that the function did not change. So on the right side, we can replace x squared plus6 with f of X on the left side. We still have F of negative x so whenever F of negative x is equal to a positive f. Then it's an even function so if you

**How To Tell If A Function Is Even Or Odd**sea number. It's even thought of that number as being multiplied times X is 0 and 0is an. Even number like 2 4 6 what about cubed minus 8x is that even or is it odd well. When it be seen X is basically to the first power now 1 & 3 are odd numbers. So therefore this is going to be an odd function now it's true. Let's find F of negative x this is negative x to the 3rd power minus 8times negative x negative x to the third power. Negative x cubed negative 8times negative x is positive 8x so now notice. All signs change so to verify that is odd take out a negative 1ife. We factor out a negative 1 it's going to be positive x cubed minus 8xand. We can see X cube minus8x is the same as f of X. Therefore F of negative x is equal to off X which means.

It's an odd function now what about this. Example To the
third minus five x squared plus two is it. Even or is it odd to notice that 3
is an odd exponent 2 is an even exponent. Whenever you see even and odd
exponents together. You know that it's going to be neither it's not even or odd
so that ‘show you could say. I'm going to snipe it but let's prove it so let's plug in negative x negative x to the third powers. Negative x cubed and negative
x tithe second power it's positive x squared. So this is what we have now let's
check to see if it's an even function right. Now the original function should be
the same as a new function but notice. It's not the same the sign for X cubed
change with the sign for x squared + 4 - did not change. Therefore it's not the
same as the original function. We can make the statement as those negative x
does not equal f of X it's not. Even now to check to see if it's an odd
function. We need to take out or factor out a negative 1so. All the signs will
change negative cubed will become positive x cubed negative 5 x. Squared will
become positive** How To Tell If A Function
Is Even Or Odd** 5 x squared + 2 will become negative 2. Now, do we
have the same function as the original function notice? These two are not the
same-cubed looks very similar. However negative five x squared is not the same
as five x squared. Therefore we say that F of negative x does not equal
negative f of X which means. It’s not an odd function so if it's not even and
if it's not odd then by default it has to be neither. How you can prove if it's
neither now let's spend a few moments talking about graphs. Even function will
be symmetric about the y-axis. An odd function is symmetric about the origin.
If it's not symmetric about the origin or the y-axis then it's neither so let's
take a look at x squared.

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