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in this episode you’ll learn what an even function is and what an odd

function is oh and by the way does every function have to be even or odd

and why is this important at all so let’s do some math [Music]

all right we’re going to begin by talking what an even function is first

so function is called even if it absorbs negatives so the way this

episode is constructed or structured is we’re first going to talk about even and

odd functions and we’re going to get an idea of what those are

and then we’re going to talk about how to apply those and why they’re important

so in the first way i’m going to go with even first right so

um function is even if if it absorbs the negative sign

so let’s look at something like this right here is this an even function

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right here so to to check that i’m going to substitute in a minus x

so this is what happens when you substitute in an x if you substitute in

a minus x then this right here gets a minus x and then that’s all squared

and then plus two and then now what does the uh squared do

that’s minus x times minus x so this is just x squared plus two

but this is just exactly what the original is so

what we shown is that f of minus x is actually equal to the original and so

therefore we can say so this means therefore f is an even function

and so now let’s see what happens when we try

something a little bit different what if you try x squared plus three

so now i’ll try that again so i’m going to substitute in a minus x

so i substitute a minus x here so minus x squared and this time we have

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plus three so this is x x squared plus three

and that’s what the original function is right here

so this is f of x and so yeah this second f is also a function

is an even function so both of them are even functions so

it’s not really the two or the three that matter it’s the power of the x that

is going to be handling the inputs so both of these are examples of even

functions right here so now let’s look at what an odd

function is before we go any further i like to contrast and compare so here’s

the definition of an odd function let me get rid of some of this right here

so an odd function is you substitute in a minus x and instead of getting out the

original function now we’re going to get minus times the

original function so that means it’s odd so for example

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what if i look at x to the third plus 1 is this an odd function well let’s try

and see what happens when i substitute in a minus x

so now i’m going to get minus x to the third plus one

so this is minus x times minus x times minus x

which is just minus x to the third and then plus one

now that is not the same as this one right here

if we had minus x to the third minus one then i could factor out a minus times

both of them and then that would be minus f of x

but we don’t have that we don’t have minus x to the third minus one

we have minus x to the third plus one and this right here is not equal to

minus f of x right so this right here is not an odd function

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the plus 1 right here ruined it so what happens if i do a plus two minus x

and now we have minus x to the third plus two

and then it’ll be minus x to the third plus two

so it’s not the one it’s not the two you know these these are not odd functions

but what about if we didn’t have a number here so what about if we just

looked at this function right here x to the third

is this an even or an odd function well if i substitute in a minus x here for

this one then i’m going to get minus x to the third

which is minus x to the third which is minus f of x right f of x is x

to the third so i have a minus f of x so

now this right here is an odd function f is an odd is anode function

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so when you substitute in minus x’s how does the function behave or how does it

act or interact and this one right here is odd these two are neither even or odd

and the first two examples we saw evens all right so now we know the answer to

the question that i asked at the beginning does every function have to be

even or does it have to be an odd and the answer is no these two right here

are not even and they’re certainly not odd all right so um here’s the you know

definitions of both even and odd even means it absorbs the negative it’s

like it’s like as if you never plugged it in

and then odd means you can pull out a negative of the whole function

and so even or not and so let’s look at um some more examples here

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okay so let’s look at this function right here f of x equals x to the sixth

minus two x squared plus three so even or odd or neither

so let’s plug in a minus x so i’m gonna get minus x to the sixth

minus two times minus x squared plus three so notice wherever there was

an x i substituted in a minus x so x to minus x

x to minus x and now we simplify so what’s minus x to the sixth right so

it’s an even number of x’s of minus x’s sorry so it’s all going to be en end up

being positive and here’s an even number of minus x’s

so it’s minus x times minus x so the negatives have multiplied to a positive

and n plus three and so this is back to the exact original function

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so when we plugged in a minus x we got back out the original

so this is an even function is an even function

all right very good so let’s look at another one so let’s look at let’s call

this one here g of x so let’s say g of x is x to the third minus 5x

so is this one even or is it odd or is it neither

let’s see let’s plug in a minus x and see how it behaves

so this will be minus x to the third minus five times minus x

so everywhere there was an x i put a minus x and now let’s simplify

so minus x times minus x times minus x is a minus x to the third

and this will be negative times a negative so this will be 5x and

that does not look like the original does it so this is a minus 5x and this is a

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plus 5x however can we rewrite it can we write it like this minus and then

a minus 5x and then now what i’ll do is i’ll factor

out a minus from both of these so this would be minus and then x to the third

and then the minus gets factored out and so i have a minus 5x left here

and this is back to the original isn’t it so this is minus g of x here

and so yes this is an odd function so i’ll say is and

actually i like to put that below is an odd function

all right there we go there’s two examples one of them is even and one of

them is odd so let’s look at one more perhaps

so here we go let’s erase this real quick let’s use a different variable here

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let’s call this one here f of t and let’s say we have t squared plus 2t minus 3.

and the question is is this an even or an odd function or neither

so let’s try let’s substitute in a minus t so i have minus t squared

plus two times minus t minus three and so this all is minus t times minus t

which is t squared and then this is a minus 2t and then this is minus 3.

now does that look like the original and the answer is no

all of them are the same except the middle one here

and so there’s no way that we’re going to be able to

factor out a minus sign or whatever because if you factor out a minus sign

you’re going to have to do it for you know multiple of them so it’s not

going to work so i’m going to say neither even nor odd

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and so there’s an example of one that’s not neither positive nor negative

all right so uh now let’s see why this even and odd business is even important

so here we go let me get rid of this uh stuff right here real quick

all right so a function is said to be even

right remember what the word here even means

you want to have it in your mind f of minus x equals f of x that’s what even

means if you plug in a minus x you simplify it you get back to the original

function so a function is said to be even if its

graph is symmetric with respect to the y axis

so what does it mean to be symmetric with respect to the y axis well let me

give you an example right here so this is a graphical representation of

the function right symmetric with respect to the y axis and this is an algebraic

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test to check so even you can do either approach if

you have some graphical information you can check that way or if you have an

actual rule for your function then you can actually check this right here so

these two things are going to be synonymous though just a different approach

so let’s look at a graph of a function that looks like this here

so it’s going to come and it’s going to loop down and come back up

and then it’s going to bounce and then do the same thing again here and then

come back up here and so this function is going to be f of x equals

x to the fourth minus 4x squared and we’re going to have a point here and

a point here and this is going to be the point here uh 2.2 um well

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let’s go this one right here zero so this is the same height here which is um

about fourish and this is zero here and this is the point here one minus three

and this is the point here minus one minus three

so these two have the same height here all right and so this graph here is

symmetric with respect to the x-axis or sorry with respect to the y-axis so

we’re going to say symmetric and let’s abbreviate this with respect to

the y-axis okay so this part right here looks

exactly the same as this part right here so this part right here is

minus two zero and this is two zero all right and these are the same height here

um and whatever this x value is here it’s about 2.2

this is minus 2.2 and then they have the same height which is about 4.

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all right so this is symmetric with respect to the y

axis now let’s check to see if this function is even so i’m going to put in

a minus x here and i’m going to get minus x to the fourth

and then minus 4 and then minus x and then a squared

right and so this will be minus x times minus x times minus x times minus x we

have four of them right so it’s x to the fourth minus four

and then this right here is just an x squared

and that is the exact original function right here so when we substitute a minus

x the minuses get all absorbed and we get back the original function right

there so this is an even function and so now we can kind of understand

better what the sentence right here is saying a function is said to be even if

it’s graph is symmetric with respect to the y-axis

so anytime you take a point over here on the graph any point right here say x y

[Music] this point over here will automatically

be on it also and what will this point right here be it will be minus x y

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it’ll have the exact same high but this will be an x distance and this will be a

minus x distance so that means symmetric with respect to the y axis

if x y is on the graph then minus x y has to be on the graph it’s symmetric

with respect to the y axis in other words the y’s are the same

all right and so there’s the intuitive idea this is a graphical approach if you

see symmetrical with respect to the y-axis then you know it’s an even function

automatically if you know it’s an even function automatically then the y’s will

be the same here but you’ll have different x’s you’ll

have a positive x you’ll have a negative x

so then you know it has to be symmetric with respect to the y axis

all right very good so what about odd functions now so for odd functions

it’s going to be a little bit different so um

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let’s look at one that we said earlier that was an odd function it was

x to the third so this will be f of x equals x to the third

so this is symmetric with respect to the origin and what does that mean

if x y is on the graph then negative x negative y is on the graph right there

so whatever this point is for example this is the point here for example 2 8

right 2 to the third is 8. then this would be the point here minus 2 minus 8.

so any point you pick on the graph x y minus x minus y so this is cement

symmetry with respect to the origin symmetry with respect to the origin

and is this an odd function well we already showed that that’s an odd function

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um earlier in this episode so maybe let’s look at a more

interesting example besides just x to the third

let’s look at something that looks like this so this time it comes down and

comes up and then it comes up and then and then it goes down

so this height right here is 1 4 and this height right here is minus one

minus four and this is the point right here minus three three

and this is the point right here [Music] three minus three and so

we’re going to be looking at this is symmetric with respect to the origin and um

so this is an odd so this right here the graph represents an odd function

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so whatever the equation is for this we know it has to satisfy

minus x equals minus f of x so once you can see that it’s symmetric

with respect to the origin then you know it has to be an odd

function it has to satisfy this so you can tell that by looking at the graph

so even if you don’t have an actual explicit equation for it you know that

it has to satisfy this it has to be an odd function because it’s symmetric with

respect to the origin for any x that’s on there minus x minus y is also on the

graph if i take this one right here then the negative of this which is 1 and the

negative of this right here is also on the graph so

that is the connection between odd and the symmetry and so

by looking if something is even or odd you can more easily make the graph or

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conversely if you have a graph then you can tell if it’s an odd function or even

function or something like that so odd and even also come up

also come up when you start looking at trigonometric functions all right so

there we go that’s it for this episode right here i hope you enjoyed it

i look forward to reading your comments below and i’ll see you in the next

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