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Even, Odd, or Neither Functions The Easy Way! - Graphs & Algebraically, Properties & Symmetry

in this video we're going to talk about

how to sell if a function is even odd or

neither we're going to talk about how to

do it the easy way and also how to do it

the way that your teacher wants you to

do it so let's begin the first thing you

need to know is how to tell if it's even

a 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 and then it's even

now what about if it's odd it's odd if F

of negative x is equal to negative f of

X so that is if you replace negative x

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 of X

so what does that mean well let's say if

you replace X with negative x and some

signs change while others don't then

it's going to be neither also if it

doesn't equal f of x2 it's also neither

so it can't equal negative f of X or f

of X so basically as long as is not even

or odd

it's neither so let's start with our

first example let's say that f of X is

equal to X to the fourth plus 3x squared

so is 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 are 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 x

squared now what is negative x raised to

the fourth power that's basically

negative x times itself 4 times negative

x squared is 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

a positive result 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

X 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 help

us determine if it's even or odd look at

the exponent five is it even or odd five

is an odd number and three is also an

odd number since all of the exponents

are odd the function is going to be an

odd function and that was proven 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 x

times negative x is positive x squared

times another negative x that's negative

x cubed so this is going to be negative

x 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 in order to prove that this

function is an odd function what you

want to do is you want to factor 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 X

cube will become positive two x cube now

notice that this portion inside the

brackets X to the fifth plus two X cube

is equal to the original function so at

that point what you want to do is

replace it with the original function f

of X so therefore we can say that F of

negative x is equal to negative f of X

which is the definition of an odd

function and that's how you can prove it

now what about this one let's say that f

of X is equal to x squared plus 6 is it

even or odd well you know x squared is

an even component because it has an even

exponent what about 6 well 6 is the same

as 6 X to the 0 anything raised to the 0

power is 1 so X to the 0 is 1 which 6

times 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 x

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 plus

6 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 of X

then it's an even function so if you see

a number it's even think of that number

as being multiplied times X is 0 and 0

is an even number like 2 4 6 what about

X cubed minus 8x is that even or is it

odd well when it be seen X is basically

X to the first power now 1 & 3 are odd

numbers so therefore this is going to be

an odd function now it's true that

so let's find F of negative x this is

negative x to the 3rd power minus 8

times negative x negative x to the third

power is negative x cubed negative 8

times negative x is positive 8x so now

notice that all signs change so to

verify that is odd take out a negative 1

if we factor out a negative 1 it's going

to be positive x cubed minus 8x

and as we can clearly see X cube minus

8x is basically the same as f of X so

therefore F of negative x is equal to f

of X which means that it's an odd

function now what about this example X

to the third minus five x squared plus

two is it even or is it odd so 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's

how 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 power

is negative x cubed and negative x to

the 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 if it's

an even function right now the original

function should be the same as a new

function but notice that it's not the

same

the sign for X cubed change with the

sign for x squared + 4 - did not change

so therefore it's not the same as the

original function so 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 1

so all the signs will change negative x

cubed will become positive x cubed

negative 5 x squared will become

positive 5 x squared + 2 will become

negative 2 now do we have the same

function as the original function notice

that these two are not the same

x-cubed looks very similar however

negative five x squared is not the same

as five x squared so therefore we say

that F of negative x does not equal

negative f of X which means that 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 so that's how you can

prove if it's neither now let's spend a

few moments talking about graphs an even

function will be symmetric about the

y-axis an odd function is symmetric

about the origin and if it's not

symmetric about the origin or about the

y-axis then it's neither so let's take a

look at x squared because it has an even

exponent we know it's an even function

the graph of x squared looks like this

it's basically a you let's do that again

it's an upward you notice that there's

symmetry about the y axis so that means

that it's an even function now if you

have a constant let's say like 3 that's

an even function f of X is the same as Y

by the way if you were to plot y equals

3 it's going to be a horizontal line at

3 and notice that this line is symmetric

about the y axis the left side looks

exactly the same as the right side so

therefore a constant by definition has

even properties now what about the graph

X cubed or y equals x cubed we know it's

an odd function this graph is an

increase in function it looks like this

as you can see there's symmetry about

the origin this side when quadrant 1

looks exactly the same or it looks like

a mirror image of the other side and

quadrant drink

so that's an example of an odd function

it's symmetric about the origin

and then there's the graph f of x equals

x or y equals x to the first power

one is an odd number but let's see why

this function is odd using a graph so y

equals x is basically a line that

increases at a 45 degree angle and as

you can see it's symmetric about the

origin

the side in quadrant one that looks like

the same as the one in quadrant three so

there's symmetry about the origin which

is the property of an odd function how

would you describe a function that looks

like this is it even odd or neither now

here there's no symmetry about the x I

mean about the Y axis or the origin so

this is neither

now what about this one is it even odd

or neither notice that the left side

looks the same as the right side so

therefore is symmetric about the y-axis

which means that it's an even function

here's another example for you determine

if this one is even or if it's odd or if

it's neither now it's not drawn

perfectly to scale so use your good

judgment so what would you say is it

even odd or neither it is symmetrical

about the y-axis the origin or neither

it's not symmetric about the y-axis the

right side does not look the same as the

left side however there is symmetry

about the origin

notice that quadrant one looks similar

to quadrant dream now this blue line

could keep on going down so even though

the height doesn't seem the same it can

keep going in that direction and notice

that quadrant four looks like a

reflection of quadrant two as you can

see the symmetry about the origin

which makes this function and odd

function is another one for you

is this an even or is it an odd function

so notice that the right side looks the

same as the left side therefore it's an

even function it's symmetric about the

y-axis what about this example

is it even odd or is it neither well we

know it's not even the right side does

not look the same as website and it's

not odd you can clearly see a difference

between this section and this section

it's not symmetric about the origin

because quadrants of four and Quadrant

two doesn't have the symmetry about the

origin they don't look the same so this

case this function will be neither

it's neither even nor is it odd let's

try one more example now what if we have

let's say a circle let me draw a better

circle it's not perfect but let's say

it's a well rounded circle is it even

odd or neither

well the circle does have even

properties as you can see the right side

looks the same as the left side this

whole side did the same so there is

symmetry about the y-axis

now what about about the origin is it

symmetric about the origin what would

you say notice that the side in Quadrant

1 looks like a reflection as the

Learning quadrant dream so there's some

symmetry about the origin in Quadrant 2

and 4 are symmetrical about the origin

so this graph is symmetric about the

y-axis and about the origin as well so

that is does that make it even or odd

now technically speaking we can't really

say it's even or odd

because it's not a function this

function doesn't pass the vertical line

test so we can't say it's an odd

function it's not an even function maybe

it's neither because it's not a function

so think about that one