right over here I have the graph of f of X and what I want to think about in this
video is whether we could have sketched this graph just by looking at the
definition of our function which is defined as a rational as a rational
expression we have 2x plus 10 over 5x minus 15 so there's a couple of ways to
do this first you might just want to pick out any numbers that are really
easy to calculate so for example what happens when X is equal to zero so we
could say F of 0 F of 0 is going to be equal to well all the X terms are going
to be 0 so you're going to be left with 10 over negative 15 which is negative 10
15 which is negative 2/3 so you could plot that when x equals 0 f of x or y
equals f of X is negative 2/3 and we see that point let me do that in a darker
color you see that point right over there so we could have plotted that
point we could have also said when does this function equal zero well the
function is equal to zero when the only way to get the function equal to zero is
if you get this numerator equal to zero so you could try to solve 2x plus 10 is
equal to zero well that's going to happen when 2x is equal to negative 10 I
just subtracted 10 from both sides and if I divide both sides by 2 that's going
to happen when X is equal to negative 5 and you see that you see this right over
here when X is equal to negative 5 the function intersects the x-axis so that's
just two points but that still doesn't give us enough to really form this
interesting shape over here and you could think about what other functions
have this type of this type of shape so now what I want to think about is the
behavior of the function at different points so first I want to think about
when this function is undefined and what type of behavior we might expect for
that function when it's undefined so this function is going to be undefined
the only way I can think of to make this undefined is if I make the denominator
equal to zero we don't know what it means to divide by zero that is
undefined so this the function is going to be undefined when five x this in blue
when five X minus 15 is equal to zero or adding 15 to both sides when five X is
equal to 15 or dividing both sides by five
five when X is equal to three f is undefined now there's a couple of ways
for a function to be undefined at a point you could have something like this
so let's let me draw some axes right over here so let's say that this is
three you could have your function it could look like this it could be defined
it might kind of approach something but just not be defined right at three and
just keep on going like that or the other possibility is it might have a
vertical asymptote there and if it has a vertical asymptote it's going to look
something like this it might it might be approach it might just pop up to
infinity and then it might pop down from infinity on this side or it might go
from negative infinity right over here that says that's what a vertical
asymptote would look like that as we approach from the left the graph is
approaching a vertical but it never it never quite gets two x equals three I
guess one we could say it or the function is defined not defined at x
equals three and as you approach from the right the same thing the function
just in this case drops down it almost becomes vertical its approaching
negative infinity as X approaches three from the positive direction so how would
we know obviously when you look at here when you look we know the graph ahead of
time and if you say okay this is e this is x equals ten so let's see how many
this is one two three four five so each of these are two so x equals three is
right over here when you look at the graph if you had
the graph in front of you you'd see oh look this is indeed a vertical asymptote
just looking at the graph you see that you have a vertical asymptote at x
equals three and let me write that down vertical asymptotes vertical asymptotes
asymptote at x equals three at x equals three but how would you have known that
how would you have known that if you didn't have the graph here if you just
had this we know it's not defined at three but how do we know it's not a
point discontinuity and not a instead of a vertical asymptote well there's a
couple of ways to do it one ways you could try values near three and see what
happens so for example you could get your
calculator out and you could try let's say three point zero one so if you say
two times three point zero one plus ten X G well that's the numerator and then
I'm going to divide that the numerator by five times three point zero one minus
fifteen it gets us a fairly large number it's kind of exploding on us and if
we've got even closer so if we did two times three point zero zero one plus let
me plus ten plus ten divided by five times three point zero zero one so now
I'm just now I'm trying X is equal to three point zero zero one minus 15 minus
15 we see we get even a larger number so as X gets closer and closer to three f
of X seems to be exploding it seems to be approaching positive infinity so
that's one way to say okay this looks like at least from this side we are
approaching we are approaching positive infinity so we would have been able to
draw we would have been able to draw something like that and then you could
have tried values below you could have tried values below so you could have
said I actually let me just let me just put the last entry here and let me just
change the 3.00 ones to two point nine nine nine two point nine nine nine and
whoops let me go over here we have two point nine nine nine and we get notice
we're going really negative now we're approaching negative infinity so if you
just try that out that would give you a pretty good indication that the graph is
looking something like that right over here which also seems to match
connecting these two points that we've already thought about but now let's see
what's going on as X approaches really large values or really negative really
positive values are really negative values it looks like there is a
horizontal asymptote here just looking at the graph just looking at the graph
it seems like there's some value that as X approaches really large values really
positive values f of X is going to be approaching
that value that asymptotes from above and as X becomes really negative it
looks like f of X is approaching that from below but how would we be able to
figure that out just by looking at this well one thought experiment is just to
say well what happens to f of X as X approaches infinity so let me write that
down as X approaches infinity then then f of X is going to approach what well as
X approaches larger and larger values the positive 10 and the negative 15
start to matter a lot less that the highest degree term in the numerator
denominator start to dominate so we could say as X approaches infinity f of
X is getting closer and closer to 2 x over 5 X which is which is 2/5 which is
equal to 2/5 so you could say f of X is approaching 2/5 and if you really want
to kind of see that a little bit more a little bit more concretely let's imagine
different values for X as X gets larger and larger and larger so if we have so X
f of X if X is 1 that f of X is just going to be 2 plus 10 over 5 minus 15 so
here the 10 and the negative and the subtracting the 15 matter a lot but if X
was if X were a thousand then f of X would be 2,000 plus 10 over 5,000 minus
15 so now the 2,000 and 5,000 are really kind of setting the agenda and then if X
were if X were let's say 1 million 1 million and I just used blue for a more
contrast then f of X would be 2 million 2 million plus 10 we move over to the
right a little bit 2 million plus 10 over 5 million 5 million minus 15 so
here the 10 and the 15 are almost inconsequential you could imagine if X
were a billion or trillion or googol then these then the 10 and the negative
15 starting to matter a lot and lot less so as X approaches infinity these matter
less the higher degree terms the degree terms matter so f of X is going
to approach 2x over 5x which is 2/5 so f of X is approaching 2/5 and that's what
this line looks like 2/5 is the same thing as 0.4 so f of X and we see that
in the graph f of X is approaching that but it's not quite getting close to it
it's it was it's not quite getting there it's getting closer and closer to it as
X goes to infinity but it's not quite getting there because you're always
going to have that plus 10 and that minus 15 there's you're never going to
be exactly at 2/5 and the same thing is happening as X gets more and more and
more negative you can make all of these negative values so if if this were
negative 1 then it would be negative 2 negative 5 this was negative 1000 would
be negative 2000 over negative 5000 if there were negative 1 million it would
be negative 2 million over negative 5 million but you see even in this case f
of X is approaching 2x over 5x which is approaching 2/5 or you could say it's
approaching negative 2 over negative 5 which is still 2/5 and you see that
right over here so we would say that this function has a horizontal asymptote
horizontal horizontal asymptote asymptote at y equals the horizontal
line right over here is y is equal to 2/5 so hopefully this graph here is
helping us appreciate what these vertical and horizontal asymptotes
actually are but if we didn't have the graph we could have said okay we're
undefined at x equals 3 we could test some values around it and so we could
say okay look it does look like we're approaching negative infinity as X
approaches negative 3 from the left it looks like we're approaching positive
infinity as X approaches negative 3 from the right so we could we could put we
could draw that kind of blue point right there we could graph these two points
when does F equal 0 what happens to F when x equals 0 and
then we could think about the behavior as X approaches infinity or negative
infinity as X approaches negative as X approaches infinity or negative infinity
and draw this horizontal asymptote and between all of those that would have
been a pretty good way to be able to sketch this actual graph