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Let's do some problems with plotting points and graphical

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representations of functions.

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So I didn't write the problem here, but what they ask us to

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do here is figure out the coordinates of

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each of these points.

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So let's just do that.

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So first we have this point a right there.

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So I'll just write a is at the coordinate.

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So we write the x-coordinate first. So its x-coordinate is

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how far to the left or the right it is of the origin.

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And it is 1, 2, 3, 4, 5, 6 to the left, or it is negative 6

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of the origin, or it's at the coordinate negative 6.

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So negative 6, that's right there.

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And then its y-coordinate, which is how high it is, that

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is right there.

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That's 4.

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Negative 6, 4.

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Let's do b right there.

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b is at the coordinate-- let's see, its x-coordinate, and

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just drop down there, is 7.

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And its y-coordinate, how high it is, is 6.

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All right.

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Let's do c.

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The x-coordinate, and you can read that, it's negative 8.

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It's 8 to the left of the origin.

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Negative 8.

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And its y-coordinate, it's 2 below the origin.

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So its y-coordinate right there is negative 2.

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This is, I think, not too painful.

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Part d, or coordinate d, or point d.

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Point d, its y-coordinate, it's at 4.

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And then its-- I'm sorry, its x-coordinate is at 4, and then

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its y-coordinate, how far down it is, or in this vertical

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axis, it is-- this looks like negative 7.

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And then, finally, we are on e.

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I'm picking out a nice color for e.

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e right there.

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Its x-coordinate is 5.

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It's right on the x-axis at x is equal to 5.

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And its y-coordinate-- well, it isn't above or below the

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origin or the x-axis, so its y-coordinate is 0.

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You could draw a line and go straight there to 0.

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So there we go.

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We figured out all of those coordinates.

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Now this problem 5 here-- let me scroll over a little bit--

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they say determine whether each relation is a function.

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So here the trick is to realize that a relation is not

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a function if they define two values for a given x.

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Let me give an example here.

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So if I wrote that f of x is equal to 5 if x is equal to 1,

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or it's equal to 6 if x is equal to 1.

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This makes no sense.

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Why doesn't it make any sense?

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Because if I put a 1 in there, I don't know what

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I'm going to produce?

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Am I going to produce a 5?

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Am I going to produce a 6?

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This is a badly structured function.

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This is not a function.

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So if there's any situation for the same input value, they

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define two different output values, we're not going to

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have a function.

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So let's see if they do that here.

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So in this first part, part a, they say, if you could imagine

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if x is 1 then y is 7.

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If x is 2, y is 7 as well.

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That is OK.

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You could have two x values getting the same y value.

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For example, that'd be like saying f of x is equal to 7 if

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x is equal to 1 or 2.

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This is completely fine.

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For two different x values you can get the same output value,

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but you can't have two different x values giving the

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same-- sorry-- you can't have the same x value producing two

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different outputs.

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Because then you don't know, hey, if you went f of 1, you

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don't know what f of 1 is.

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f of 1, is it a 5, is it 6?

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You don't know.

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Here you know what f of 1 is, it's 7.

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Here you know what f of 2 is, it's 7.

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So, so far so good.

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So when you have 2, you have a 7.

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When your input is 3, you get an 8.

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When your input is 4, you get an 8.

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So, for example, our function definition, so it's 8 if x is

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equal to 3 or 4.

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And then 5.

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And then our function is equal to 9 if x is equal to 5.

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So part a, this is our function definition, right

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here, for part a, which is a completely legitimate function

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definition.

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You give me any value 1, 2, 3, 4, or 5, which would be the

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domain in this situation, and I will tell you what the value

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of that function is at any of those points.

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And the range would be 7, 8, or 9.

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So part a is definitely a function.

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Now part b.

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Let's see, if x is 1, y is 1.

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But then they say, if x is 1, y is negative 1.

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Well, that makes no sense.

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They're doing this right here.

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They're trying to make a function, where they say this

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function is going to be equal to 1 if x is equal to 1, but

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then it's going to be equal to negative 1 if x is equal to 1.

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So if I were to take f of 1, I don't know what

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it's going to be.

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Is it 1 or negative 1?

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I don't know, do I take this 1 or do I take that 1?

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So this is not a function.

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So part b is not a function.

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That relation is not a function.

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All right.

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Let's do a couple more.

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Problem 6.

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Write the function rule for each graph.

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So we have this little v looking thing.

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So we could write it a couple of ways.

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Let's call it f of x.

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And you could call it g of x, or h of x, but if you haven't

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used your f yet, people tend to use f of x.

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So this is x.

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So let's see, it looks like it's one line when x is

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greater than the 0, and another line when x

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is less than 0.

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So it's one thing when x is, let's say, greater than or

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equal to 0.

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And another thing when x is less than 0.

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And I'm going to merge the two in a second.

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So what does the line look like here?

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When x is 0, y is 0.

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When x is 2, y is 1.

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When x is 4, y is 2.

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It looks like no matter what x is, y is going

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to be 1/2 of that.

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When x is 6, y is 3.

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So it's equal to 1/2x, when x is greater than 0.

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And then when x is less than 0, when x is

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negative 2, y is 1.

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When x is negative 4, y is 2.

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So here it looks like it's the negative 1/2 of it.

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Negative 1/2 times negative 4 is positive 2.

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So it's negative 1/2 times x, when x is less than 0.

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So this is a completely legitimate answer.

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But if we wanted to make it a little bit simpler, or clean

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it up a little bit, we could write this function definition

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as f of x being equal to-- instead of dividing it between

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greater than or less than 0, let's just take the absolute

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value of x and then multiply that times 1/2.

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Because here, that obviously works for positive values,

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because the absolute value of x will be equal to x.

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But then for negative values, the absolute value of x is

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equal to negative x.

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So you take negative 2 here.

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Negative 2, take the absolute value, you get 2

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times 1/2 is 1.

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So either of these would be legitimate function

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definitions.

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Problem 10.

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Use the vertical line test-- let me switch colors for this

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one-- use the vertical line test to determine whether each

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relation is a function.

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Now, the vertical line test is just a visual way of doing

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exactly what we did in this problem over here.

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Something is only a function if for a given x value, you

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only have one y value.

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So for example, when they say a vertical line, that means at

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any point I should be able to draw a vertical line on this

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function and only intersect it once.

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Because a vertical line says when x is equal to 3, there's

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only one value that I can get on our function.

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That's what I'm doing with a vertical line.

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When x is equal to 0, there's only one point on the function

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that is mapped from x is equal to 0.

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So this one right here is a function.

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Any vertical line you draw will only intersect the

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function once.

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This one, very clearly, you draw any vertical line.

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You draw any vertical line here and you're going to

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intersect the graph twice.

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So this is saying, this vertical line that I just

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drew, this is essentially saying f of 2 could be this

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point over here.

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This looks like, I don't know, maybe it's equal to, it could

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be equal to 1.9 or it could be-- what is

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this?-- negative 1.9.

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This is not a proper function definition.

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I don't know whether f of 2 is this value or that value.

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I don't know whether f of 0 is that value or that value.

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And you could keep going across the whole number line.

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You don't know whether f of 5 is this value or this value.

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So this is not a function.

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This relation is not a function.

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And it's the same logic as we did before, but we're saying

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the vertical line.

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You could draw a vertical line and you can intersect the

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relation or the graph twice, so it's not a function.