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