What Are Even and Odd Functions (and Why Is This Useful?)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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.

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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
episode if you enjoyed this video please like and subscribe to my channel
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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