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In this video, I want to do a few examples dealing with functions. Functions tend to be something that a lot of students find difficult, but I think if you really get what we're talking about, you'll see that it's actually a pretty straightforward idea. And you sometimes wonder, well what was all of the hubbub about? All a function is, is an association between two variables. So if we say that y is equal to a function of x, all that means is, you give me an x. You can imagine this function is kind of eating up this x. You pop an x into this function. This function is just a set of rules. It's going to say, oh, with that x, I associate some value y. You can imagine it is some type of a box. That is a function. When I give it some number x, it'll give me some other number y. This might seem a little abstract. What are these x's and y's? Maybe I have a function-- let me make it like this. Let's say I have a function definition that looks like this. For any x you give me, I'm going to produce 1 if x is equal to-- I don't know-- 0. I'm going to produce 2 if x is equal to 1. And I'm going to produce 3 otherwise. So now we've defined what's going on inside of the box. So let's draw the box around it. This is our box. This is just an arbitrary function definition, but hopefully it'll help you understand what's actually going on with a function. So now if I make x is equal to-- if I pick x is equal to 7, now what is f of x going to be equal to? What is f of 7 going to be equal to? So I take 7 into the box. You could view it as some type of a computer. The computer looks at that x and then looks at its rules. It says, OK. x is 7. Well x isn't 0. x isn't 1. I go to the otherwise situation. So I'm going to pop out a 3. So f of 7 is equal to 3. So we write f of 7 is equal to 3. Where f is the name of this function, this rule system, or this association, this mapping, whatever you want to call it. When you give it a 7, it'll produce a 3. When you give f a 7, it'll produce a 3. What is f of 2? Well, that means instead of x is equal to 7, I'm going to give it an x equal 2. Then the little computer inside the function is going to say, OK, let's see, when x is equal to 2. No, I'm still in the otherwise situation. x isn't 0 or 1. So once again f of x is equal to 3. So, this is f of 2 is also equal to 3. Now what happens if x is now equal to 1? Well then it's just going to turn over this. So f of 1. It's going to look at its rules right here. Oh look, x is equal to 1. I can use my rule right here. So when x is equal to 1, I spit out a 2. So f of 1 is going to be equal to 2. I spit out f of 1, which is equal to 2 in that situation. That's all a function is. Now, with that in mind, let's do some of these example problems. They tell us for each of the following functions, evaluate these different functions-- these are the different boxes they've created-- at these different points. Let's do part a first. They're defining the box. f of x is equal to negative 2x plus 3. They want to know what happens when f is equal to negative 3. Well f is equal to negative 3, this is telling me what do I do with the x? What do I produce? Wherever I see an x, I replace it with the negative 3. So it's going to be equal to negative 2. Let me do it this way, so you see exactly what I'm doing. That negative 3, I'll do it in that bold color. It's negative 2 times negative 3 plus 3. Notice wherever there was an x, I put the negative 3. So I know what the black box is going to produce. This is going to be equal to negative 2 times negative 3 is 6 plus 3, which is equal to 9. So f of negative 3 is equal to 9. What about f of 7? I'll do the same thing one more time. f of-- I'll do 7 in yellow-- f of 7 is going to be equal to negative 2 times 7 plus 3. So this is equal to negative 14 plus 3, which is equal to negative 11. You put in-- let me make it very clear-- you put in a 7 into our function f here and it will pop out a negative 11. That's what this just told us right there. This is the rule. This is completely analogous to what I did up here. This is the rule of our function. Let's do the next two. I won't do part b. You can do part b for fun. I'll do part c after that, just for the sake of time. Now we are at f of 0. Here I'll just do it in one color. I think you're getting the idea. f of 0. Wherever we see an x, we put a 0. So negative 2 times 0 plus 3. Well, that's just going to be a 0. So f of 0 is 3. Then one last one. f of z. They want to keep it abstract for us. Here I'll color code it. So f of z. Let me make the z in a different color. f of z. Everywhere that we saw an x, we will now replace it with a z. Negative 2. Instead of an x, we're going to put a z there. We're going to put an orange z there. Negative 2 times z plus 3. And that's our answer. f of z is negative 2z plus 3. If you imagine our box, the function f. You put in a z, you are going to get out a negative 2 times whatever that z is plus 3. That's all this is saying. It's a little bit more abstract, but same exact idea. Now let's just do part c here. Let me clear this actually. I'm running out of real estate. Let me clear all of this business. Let me clear all of this business. We can do part c. I'm skipping part b. You can work on that part. Part b. They tell us-- this is our function definition. Sorry, I said I was doing part c. This is our function definition. f of x is equal to 5 times 2 minus x over 11. So let's apply these different values of x, these different inputs into our function. So f of negative 3 is equal to 5 times 2 minus-- wherever we see an x, we put a negative 3. 2 minus negative 3 over 11. This is equal to 2 plus 3. This is equal to 5. So you get 5 times 5 over 11. That's equal to 25/11. Let's do this one. F of 7. For this second function right here, f of 7 is equal to 5 times 2 minus-- now for x we have a 7. 2 minus 7 over 11. So what is this going to be equal to? 2 minus 7 is negative 5. 5 times negative 5 is negative 25/11. Then finally, well we have two more. f of 0. That's equal to 5 times 2 minus 0 So this is just 2. 5 times 2 is 10. So this is equal to 10/11. One more. f of z. Well every time we saw an x, we're going to replace it with a z. It's equal to 5 times 2 minus z over 11. And that's our answer. We could distribute the 5. You could say this is the same thing as 10 minus 5z over 11. We could even write it in slope-intercept form. This is the same thing as minus 5/11 z plus 10/11. These are all equivalent. But that is what f of z is equal to. Now. A function, we said, if you give me any x value, I will give you an output. I will give you an f of x. So if this is our function, you give me an x, it will produce an f of x. It can only produce 1 f of x for any x. You can't have a function that could produce two possible values for an x. So you can't have a function-- this would be an invalid function definition-- f of x is equal to 3 if x is equal to 0. Or it could be equal to 4 if x is equal to 0. Because in this situation, we don't know what f of 0 is. What it's going to be equal? It says if x is equal to 0, it should be 3 or it could be-- we don't know. We don't know. We don't know. This is not a function even though it might have looked like one. So you can't have two f of x values for one x value. So let's see which of these graphs are functions. To figure that out, you could say, look at any x value here-- pick any x value-- I have exactly one f of x value. This is y is equal to f of x right here. I have exactly only one-- at that x, that is my y value here. So you could have a vertical line test, which says at any point if you draw a vertical line-- notice a vertical line is for a certain x value. That shows that I only have one y value at that point. So this is a valid function. Any time you draw a vertical line, it will only intersect the graph once. So this is a valid function. Now what about this one right here? I could draw a vertical line, let's say, at that point right there. For that x, this relation seems to have two possible f of x's. f of x could be that value or f of x could be that value. Right? We're intersecting the graph twice. So this is not a function. We're doing exactly what I described here. For a certain x, we're describing two possible y's that could be equal to f of x. So this is not a function. Over here, same thing. You draw a vertical line right there. You're intersecting the graph twice. This is not a function. You're defining two possible y values for 1 x value. Let's go to this function. It's kind of a weird looking function. Looks like a reversed check mark. But any time you draw a vertical line, you're only intersecting it once. So this is a valid function. For every x, you only have one y associated. Or only one f of x associated with it. Anyway, hopefully you found that useful.