Let's do some problems with plotting points and graphical
representations of functions.
So I didn't write the problem here, but what they ask us to
do here is figure out the coordinates of
each of these points.
So let's just do that.
So first we have this point a right there.
So I'll just write a is at the coordinate.
So we write the x-coordinate first. So its x-coordinate is
how far to the left or the right it is of the origin.
And it is 1, 2, 3, 4, 5, 6 to the left, or it is negative 6
of the origin, or it's at the coordinate negative 6.
So negative 6, that's right there.
And then its y-coordinate, which is how high it is, that
is right there.
That's 4.
Negative 6, 4.
Let's do b right there.
b is at the coordinate-- let's see, its x-coordinate, and
just drop down there, is 7.
And its y-coordinate, how high it is, is 6.
All right.
Let's do c.
The x-coordinate, and you can read that, it's negative 8.
It's 8 to the left of the origin.
Negative 8.
And its y-coordinate, it's 2 below the origin.
So its y-coordinate right there is negative 2.
This is, I think, not too painful.
Part d, or coordinate d, or point d.
Point d, its y-coordinate, it's at 4.
And then its-- I'm sorry, its x-coordinate is at 4, and then
its y-coordinate, how far down it is, or in this vertical
axis, it is-- this looks like negative 7.
And then, finally, we are on e.
I'm picking out a nice color for e.
e right there.
Its x-coordinate is 5.
It's right on the x-axis at x is equal to 5.
And its y-coordinate-- well, it isn't above or below the
origin or the x-axis, so its y-coordinate is 0.
You could draw a line and go straight there to 0.
So there we go.
We figured out all of those coordinates.
Now this problem 5 here-- let me scroll over a little bit--
they say determine whether each relation is a function.
So here the trick is to realize that a relation is not
a function if they define two values for a given x.
Let me give an example here.
So if I wrote that f of x is equal to 5 if x is equal to 1,
or it's equal to 6 if x is equal to 1.
This makes no sense.
Why doesn't it make any sense?
Because if I put a 1 in there, I don't know what
I'm going to produce?
Am I going to produce a 5?
Am I going to produce a 6?
This is a badly structured function.
This is not a function.
So if there's any situation for the same input value, they
define two different output values, we're not going to
have a function.
So let's see if they do that here.
So in this first part, part a, they say, if you could imagine
if x is 1 then y is 7.
If x is 2, y is 7 as well.
That is OK.
You could have two x values getting the same y value.
For example, that'd be like saying f of x is equal to 7 if
x is equal to 1 or 2.
This is completely fine.
For two different x values you can get the same output value,
but you can't have two different x values giving the
same-- sorry-- you can't have the same x value producing two
different outputs.
Because then you don't know, hey, if you went f of 1, you
don't know what f of 1 is.
f of 1, is it a 5, is it 6?
You don't know.
Here you know what f of 1 is, it's 7.
Here you know what f of 2 is, it's 7.
So, so far so good.
So when you have 2, you have a 7.
When your input is 3, you get an 8.
When your input is 4, you get an 8.
So, for example, our function definition, so it's 8 if x is
equal to 3 or 4.
And then 5.
And then our function is equal to 9 if x is equal to 5.
So part a, this is our function definition, right
here, for part a, which is a completely legitimate function
definition.
You give me any value 1, 2, 3, 4, or 5, which would be the
domain in this situation, and I will tell you what the value
of that function is at any of those points.
And the range would be 7, 8, or 9.
So part a is definitely a function.
Now part b.
Let's see, if x is 1, y is 1.
But then they say, if x is 1, y is negative 1.
Well, that makes no sense.
They're doing this right here.
They're trying to make a function, where they say this
function is going to be equal to 1 if x is equal to 1, but
then it's going to be equal to negative 1 if x is equal to 1.
So if I were to take f of 1, I don't know what
it's going to be.
Is it 1 or negative 1?
I don't know, do I take this 1 or do I take that 1?
So this is not a function.
So part b is not a function.
That relation is not a function.
All right.
Let's do a couple more.
Problem 6.
Write the function rule for each graph.
So we have this little v looking thing.
So we could write it a couple of ways.
Let's call it f of x.
And you could call it g of x, or h of x, but if you haven't
used your f yet, people tend to use f of x.
So this is x.
So let's see, it looks like it's one line when x is
greater than the 0, and another line when x
is less than 0.
So it's one thing when x is, let's say, greater than or
equal to 0.
And another thing when x is less than 0.
And I'm going to merge the two in a second.
So what does the line look like here?
When x is 0, y is 0.
When x is 2, y is 1.
When x is 4, y is 2.
It looks like no matter what x is, y is going
to be 1/2 of that.
When x is 6, y is 3.
So it's equal to 1/2x, when x is greater than 0.
And then when x is less than 0, when x is
negative 2, y is 1.
When x is negative 4, y is 2.
So here it looks like it's the negative 1/2 of it.
Negative 1/2 times negative 4 is positive 2.
So it's negative 1/2 times x, when x is less than 0.
So this is a completely legitimate answer.
But if we wanted to make it a little bit simpler, or clean
it up a little bit, we could write this function definition
as f of x being equal to-- instead of dividing it between
greater than or less than 0, let's just take the absolute
value of x and then multiply that times 1/2.
Because here, that obviously works for positive values,
because the absolute value of x will be equal to x.
But then for negative values, the absolute value of x is
equal to negative x.
So you take negative 2 here.
Negative 2, take the absolute value, you get 2
times 1/2 is 1.
So either of these would be legitimate function
definitions.
Problem 10.
Use the vertical line test-- let me switch colors for this
one-- use the vertical line test to determine whether each
relation is a function.
Now, the vertical line test is just a visual way of doing
exactly what we did in this problem over here.
Something is only a function if for a given x value, you
only have one y value.
So for example, when they say a vertical line, that means at
any point I should be able to draw a vertical line on this
function and only intersect it once.
Because a vertical line says when x is equal to 3, there's
only one value that I can get on our function.
That's what I'm doing with a vertical line.
When x is equal to 0, there's only one point on the function
that is mapped from x is equal to 0.
So this one right here is a function.
Any vertical line you draw will only intersect the
function once.
This one, very clearly, you draw any vertical line.
You draw any vertical line here and you're going to
intersect the graph twice.
So this is saying, this vertical line that I just
drew, this is essentially saying f of 2 could be this
point over here.
This looks like, I don't know, maybe it's equal to, it could
be equal to 1.9 or it could be-- what is
this?-- negative 1.9.
This is not a proper function definition.
I don't know whether f of 2 is this value or that value.
I don't know whether f of 0 is that value or that value.
And you could keep going across the whole number line.
You don't know whether f of 5 is this value or this value.
So this is not a function.
This relation is not a function.
And it's the same logic as we did before, but we're saying
the vertical line.
You could draw a vertical line and you can intersect the
relation or the graph twice, so it's not a function.