Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Functions are correspondence where everything in the first set goes to exactly one thing in the second set. Function notation has lots of different ways, but the most typical is F of X. Now this doesn't mean multiply, it means that this X in here is the input and the whole F of X together is the output. The input is called the domain and the output is called the range. Frequently the domain is our X value and the range is our Y value. Now that doesn't have to be the case, because X&Y are really just variables, so if we're using other variables, it always goes in alphabetical order. So the input is always the first variable and the output is always the second variable. A relation is something that's not a function, so the domain in the first set might go to multiple things in the second set or the range. Vertical line test is a method to be able to determine if something is a function or not. If we have a graph and we can draw any vertical line on the graph, we know that it passes the vertical line test and thus it is a function. And a piecewise defined function means that there's just multiple pieces that we're going to look at in a graph. So the first thing to think about domain and range. The domain is the first set. So in this case, AB and C is the domain and XY and Z is the range. And the question is, is this a function or not? So when we look at A, a goes to YA only goes to Y. So it goes to one thing. So it is a function or A the A portion passes it. B always goes to Z&C always goes to X. So the fact that AB and C always go only to one location means that this one is a function. When we look at this next, 1D always goes to M so that part's OK. But E is going to both M&N. So this is not a function because that E one element went to multiple places. One thing in the domain went to multiple places in the range. Sometimes you're given the information as a list. So remember ordered pairs. The first one is our X, the second one is our Y. So we could think of two always goes to four for this first ordered pair. So I could actually make myself a picture if I wanted to. Two goes to four, five goes to 7, three goes to 6-8 goes to six. Now the three and the 8 still only go to one number. So this one is a function. Because every element in the domain 2538 goes to one element in the range 476. We don't care that that 6 has been repeated for this. Next 10 goes to 04 goes to two, but now zero goes to three. Can zero in the domain go to more than one thing in the range and be a function? And the answer's no. So this one's not a function because that zero in the domain went to both 3 and 0 in the range. What about if it was given still in a little bit different format? Does every car have a gas mileage? Well, gas mileages might vary, but for any given car at a certain time, it's going to only have one gas mileage. Your car might get 20 miles per gallon, it might get 40 miles per gallon. It might be 41.72, but every car has exactly 1 gas mileage at any given time. So yes, this one is also a function. So every input has exactly 1 output. The input is the domain, the output is the range. Multiple ways to do the notation. F of X is our output. Well, our output typically is our Y. So we could think of F of X just equaling Y. Sometimes the notation is F; X. Remember the X is the domain that inside X is the domain. The whole F of X or the Y is the range. So if we're looking at an example of this picture, this is saying F of one or when X is one, what is Y or F of X? So if X is 1, based on our ordered pair concept, we're going to go to the right one because that's when X is 1. Then I'm going to think about what happens if I drew in a line at X equal 1. There's my X equal 1 by drawing this vertical line. What I want to know is where does this X intersect the graph? And I'm going to estimate that to be about 7 in this case. So F of one means when X is one, what is YF of X equaling two? Well, this F of X, the whole thing is the Y value. So this is saying what happens when Y is 2? What's the X value? And there may be more than one. So if we look at this graph when Y is 2, if we put in a horizontal line at Y equal 2, it looks like it intersects the original graph twice. Looks like if we're estimating, I'm going to say -4 so F of -4 equal to. Now this is really, really just an ordered pair. This is the ordered pair of when X is -4 Y is 2. Now in our graph, it intersects another time, it looks like maybe 8. So there's a second location where my Y is 2, and that's when X was 8. So 8, 2. So when X is -4 or 8, Y is 2. I'm going to have you try a couple of them using the same figure up here. So remember, when X is 3, what's our Y for F of three? When F of X = 0? This is saying when Y is 0, what is the X practice? That that's an important concept? The vertical line test. Given a few pictures, we want to see if I put in any vertical line anywhere on the graph. Pretend that these are straight any vertical line. Does it only pass the graph one time? If the answer is yes, then it is a function. If the answer is no, it's not a function, it's a relation. So here it's a relation, not a function. Here it only has to do it one time. So if we're looking way off to the right, it looks like it would pass the vertical line test. But as we get further to the left of this graph, we can see that this line would pass more than once. So it's not a function. And if it's not a function, we can call it a relation. Now every function is a relation. Not every relation is a function. This last one is a function. Now when we're talking about functions, we also need to think about domains and ranges and how to define them. So if you're given an equation F of X equal three X ^2 + 5, this is saying that any value I put in for X, I'm going to square it. I'm going to multiply by three, and I'm going to add five. So is there any number that this doesn't work for? Is there any value for XI? Can't use and the answer is no. I can put in any value of XI, can square any value of XI, can multiply it by three. I can add five. So the domain here is all real numbers, and what we usually write is negative Infinity to Infinity because that tells me every single number on a number line would work. We thought about a number line. Every single number works Well. What if it was like F of X equal two X / 3 + X? Well, in this case we know we can never divide by zero, so that denominator 3 + X can never equal 0. If I solved for that, XI can see that X could never be -3. So on a number line we'd have an open dot at -3 saying it can't be there. So in interval notation, what we would write is we'd say it's true from negative Infinity all the way to three. We're not going to include this .30 negative 3, so we're going to use a parenthesis and then a union because it's also true to the right of that number -3 to Infinity. If we wanted to include the point, we would use a bracket, but when we don't include the point, we use a parenthesis. We don't care about the domain or the numerator here because the numerator can be anything. It's the denominator that gives us something that's undefined. Thank you and have a wonderful day.