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We're back with more to say about functions and relations. Suppose we're given a relation;
that could be any subset of the plane. We might be asked to the following questions.
For example, is it a function? What is its domain? And what is its range? Let's consider
these questions for the relation shown in the graph that follows. You may recall that
one way to test whether a relation is a function is by the vertical line test. See, here we
have a picture of something drawn in the plane and we would like to know: does it pass the
vertical line test? So we take a vertical line, and we sweep it along the relation from
left to right, and we ask is there any point at which the vertical line crosses the graph
twice? Well, here we see there's always one point of intersection, so this is an example
of a function. So, is it a function? Yes. It passes the vertical
line test. Next we would like to know: what is its domain? To remind you what this question
means, we're essentially asking what are all of the valid x values corresponding to points
on this graph. For example, in this case we see that the point (-5, 1.33) is a point on
the graph. Its x coordinate is negative five. Therefore, negative five is in the domain.
If we sweep this vertical line along the graph, we look for that point of intersection...now
it's (-3,2.67). As far as the domain is concerned, all we care about is the -3. And so you can
see along the x axis this line of blue points stacking up. These are all of the x values
that we have found corresponding to points on the original graph. And it turns out for
this particular relation, every real number x corresponds to a point on the graph. Therefore
we say that the domain of this relation is the set of all real numbers. I'd like to show
you another way of thinking about the same idea. So going back to this graph...rather
than looking for points along the x axis that are above or below points on the graph, I
would like to imagine squashing the graph flat onto the x axis, so various parts of
this graph will move vertically toward the x axis, like this. And the domain of the function
is the set of x values that results. And we can see that the entire x axis has been painted
by this process. Let's see that again in reverse just to give you an idea of where we're coming
from. This parabola gets squashed flat onto the x axis, and it covers the whole axis, which
means that the domain is all real numbers. So we've confirmed for the domain is all real
numbers. What about the range? Let's investigate that using both of the techniques we just
saw. So in order to find the range we will use a horizontal line that sweeps across the
graph of the relation. Here it goes. Now, whenever this horizontal line meets the relation,
we keep track of the resulting y coordinates. For example: this point, (-6.22,5.1), and
this point, (4.22,5.1), both have a y coordinate of 5.1. Therefore 5.1 is one of the valid
y coordinates associated with this relation. 5.1 belongs in the range. And we can see here
on the y axis, 5.1 has been marked. This entire stack of red points represents the range of
our relation. Another way to think about the range of this relation is to smash the entire
relation onto the y axis. That looks like this. And we see that the entire relation seems
to get smashed into this part of the y axis, starting at -4 and going up from there. These
are the same values that we saw with the stack of red dots earlier. Therefore we say that
the range of this function is the interval from -4 to infinity. This is all real numbers
from -4 on up. Now let's consider another relation. Do you think this is a function?
How do we decide? We use the vertical line test. We sweep a vertical line across the
relation. Does this vertical line ever strike the relation in more than one point at the
same time? Yes. We see here for example, there's one point of intersection here, and another
one here. Since we have more than one point of intersection, this relation does not describe
a function. So we say this is not a function because it fails the vertical line test. But
we can still ask: what is its domain? And to answer that question, we again sweep the
vertical line across the relation looking for x coordinates that correspond to points
on the graph. So here, whenever the line meets the relation, we've leave a blue mark on the
x axis. The collection of all these blue marks is the domain of the function. In this case,
that would be the closed interval from -5 to 5. And this is how he write that. There
is another way for us to think about the domain, however. Let's take a look that right now.
Rather than sweeping a vertical line across the relation, we could also just think about
smashing the relation onto the x axis, this way. And the question is: what interval do
we get when we smash the relation in this fashion? We can see that it stretches from
-5 to 5. This is the same answer that we got before. So now for our last question: what
is the range of this relation? As with the domain, we have two ways to think about the
range of this relation. One of them is by sweeping a horizontal line across the relation,
keeping track of any y coordinates corresponding to points that we meet. In this case it looks
like we're marking the y axis from -4 to 4. So we can fill this in as our answer. what
is the range? The closed interval from -4 to 4. But of course we have the second way
to think about it, which is: we smash the relation onto the y axis like this. And when
we do that, what interval do we get? Yes, it's the closed interval from -4 to 4, again.
This confirms the answer we got previously. So here's a challenge for you. I'd like you
to consider the relation shown in the graph. Do you think this relation is a function?
Pause, think about your answer, and then keep playing. Is it a function? Yes, it is a function
because it passes the vertical line test. If you'd like you can see that here. As the
blue line sweeps across, it never hits the blue line more than once. We have a function.
So now, what is its domain? Pause the video for as long as you like, and see if you can
decide what the domain is. What is its domain? All real numbers. In interval notation, that's
the open interval from negative infinity to infinity. How'd you do? Last question: what's the range of that relation?
Here's the graph; pauses as long as you need. What did you get for the range? I got the
interval closed at zero open at infinity. If you'd like to see why, just take our relation
and smashit onto the y axis. Notice how it covers the positive y axis. That's the interval
from zero to infinity. Thanks for watching!