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.