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# Graphical symmetry of functions

Video transcript

Which of these functions is odd? And so let's remind
ourselves what it means for a
function to be odd. So I have a function--
well, they've already used f, g,
and h, so I'll use j. So function j is odd. If you evaluate j at some
value-- so let's say j of a. And if you evaluate that j at
the negative of that value, and if these two things are
the negative of each other, then my function is odd. If these two things
were the same-- if they didn't have this
negative here-- then it would be an even function. So let's see which of these
meet the criteria of being odd. So let's look at f of x. So we could pick a
particular point. So let's say when
x is equal to 2. So we get f of 2 is equal to 2. Now, what is f of negative 2? f of negative 2
looks like it is 6. f of negative 2 is equal to 6. So these aren't the
negative of each other. In order for this to
be odd, f of negative 2 would have had to be equal to
the negative of this, would have had to be
equal to negative 2. So f of x is definitely not odd. So all I have to do is
find even one case that violated this
constraint to be odd. And so I can say it's
definitely not odd. Now let's look at g of x. So I could use the same-- let's
see, when x is equal to 2, we get g of 2 is
equal to negative 7. Now let's look at
when g is negative 2. So we get g of negative 2
is also equal to negative 7. So here we have a
situation-- and it looks like that's the case for
any x we pick-- that g of x is going to be equal
to g of negative x. So g of x is equal
to g of negative x. It's symmetric
around the y-- or I should say the vertical
axis-- right over here. So g of x is even, not odd. So which of these
functions is odd? Definitely not g of x. So our last hope is h of x. Let's see if h of x seems
to meet the criteria. I'll do it in this green color. So if we take h of 1-- and we
can look at it even visually. So h of 1 gets us right
over here. h of negative 1 seems to get us an equal amount,
an equal distance, negative. So it seems to fit for 1. For 2-- well, 2
is at the x-axis. But that's definitely h of 2
is 0. h of negative 2 is 0. But those are the
negatives of each other. 0 is equal to negative 0. If we go to, say, h of 4, h
of 4 is this negative number. And h of negative 4 seems
to be a positive number of the same magnitude. So once again, this is
the negative of this. So it looks like this is
indeed an odd function. And another way to visually
spot an odd function is a function-- it's going
to go through the origin, and you could essentially
flip it over on both axes. So if you flip this, the right
half, over the left half, and then flip that over
the horizontal axis, you are going to get
this right over here. So you see here we're
going up and to the right. Here we're going to go
down and to the left. And then you curve
right over there. You curve up just like that. But the easiest way to test
it is just to do what we did, look at a given x. So for example, when x
is equal to 8, h of 8 looks like this
number right around 8. h of negative 8 looks like it's
pretty close to negative 8. So they seem to be the
negative of each other. It sounds like a car crash
just happened outside. Anyway, hopefully
you enjoyed that. Not the car crash,
the math problem.