0:00

>> [music] Welcome, in this video we'll be leaning about functions and graphs.

Â Functions are probably the most important mathematical object you'll ever learn

Â about and you probably will spend most of your mathematics courses talking about

Â functions. Functions come up all the time in the real

Â world. Let's start by talking about the

Â definition of function. A function is a correspondence between two

Â sets of elements, such that to each element in the first set there is a

Â corresponding element in the second set and there's one and only one of these

Â elements in the second set. The first set, we call the domain and the

Â second set, the ones that correspond to the first set, are called the range of our

Â function. We usually denote functions by y equals f

Â of x. The way we read this is y is a function of

Â x. In other words, we input an x into a

Â function and we get some output value, y. This is what we mean by y equals f of x or

Â y is a function of x. Let's consider example of a function.

Â My function box here is denoting a function f of x.

Â I'm going to think of a function as a magic black box, or in this case, a brown

Â box, in which you put in an input and the function gives you an output.

Â So what I mean is, we take an x, we put it in into the function and out comes the

Â value f of x. I'm going to have my student, Victor,

Â who's here with me today, help me out. Victor, say hi to everyone.

Â Victor, what number would you like to put into the function first?

Â Let's input the number 1. So we put 1 into the function and out

Â comes the number 2, so f of 1 equals 2. What number would you like to try next?

Â 3, let's try f of 3. We put 3 into our function and out comes

Â the value of 8. F of 3 equals 8.

Â Let's try another one. How about 6?

Â We plug in f of x. We put it into the function and out comes

Â the value of 5, f of 6 equals 5. Let's do one more.

Â We're going to plug in a half. We plug in half as my input into my

Â function and out comes my value of 7, f of a half equals 7.

Â Let's try one more. What would you like to try next?

Â Panda. Oh well, let's give it a try, f of panda,

Â we put panda into the function and out comes tiger.

Â F of panda equals tiger. This is my function f of x.

Â Let's go ahead and look at some of its properties.

Â So you just saw my function box, f of x. Notice, I put in several numbers.

Â I put in 1, 3, 6 and a half. Those are the numbers Victor gave me, and

Â out came the numbers 2, 8, 5 and 7. Let's ignore the panda turning into a

Â tiger for right now. We can denote these values in a table, as

Â you see on to the graph. And we can also depict it as a graph,

Â where we have the x and y-axis, and we denote each value of the function as a

Â point, with the x input and the y output. Let's look at a new function, g of x.

Â Let's try a couple of inputs to see what this new function does.

Â Victor? Let's put in 1.

Â When we put in 1 to the function, the outcome is 1.

Â Let's try another one. We plug in 2, we get, 2.

Â Let's try 3. Let's put in 3 into my function, out comes

Â the value of 3. I'm, think I see a pattern here.

Â We put in 7 into our function, what comes out?

Â We get a, another 7. One more.

Â Plug in 8. Have you figured out what's happening yet?

Â We plug in 8, outcomes a value of 8. Let's take a look at this function.

Â You noticed it seems to have a pretty strong pattern.

Â Whenever I put in a 1, I got out a 1. I put in a 2, I got out a 2.

Â Every number that I input seem to be the same number I got as an output.

Â You might guess that if I put in 100 to my function, I'd get 100 out.

Â Over here, we see we got a table of our function values that we tried.

Â We also have plotted the x and y values, the inputs and the outputs, on a Cartesian

Â coordinate system or the xy plane. Let's think about this function.

Â Do we know which function this one is? Well, you've probably guessed it.

Â This is the function g of x equals x. For every x value I put in, I get the same

Â x value out of the function and I can connect those dots from my trials with a

Â straight line. Just having five data points isn't enough

Â to determine what the function actually is, but based on this, my best guess is

Â this is probably the function g of x equals x.

Â Let's look at a new function, h of x. When we're looking at h of x this time,

Â let's pay particularly close attention to figure out if h of x is a function.

Â Remember, a function is something that for every input, you get a correspondingly

Â unique output. One and only one output for each input,

Â let's give this a try. Victor, you got an input for me?

Â Let's try 1. H of 1 give me a value of, let's see, 3.

Â H of 1 is 3. Let's try another one.

Â 2, h of 2 is. 3 again.

Â Let's try another. Plug in 3, h of 3 is.

Â Let's see, 3. H of 3 is 3.

Â Let's plug in 7 now. H of 7 is 3.

Â I think you're seeing a pattern here. Let's try one more just to be sure, h of

Â 8. When we plug in 8, what do we get?

Â Well, I'm not sure this time. 3, h of 8 is 3.

Â This function seems to have for every input, we get a value of 3.

Â Does this contradict the definition of function?

Â What do you think? It doesn't contradict it, it's actually

Â it's a function that kind of seems like it's not.

Â Let's see why? The function we just saw h of x, gave us

Â an output of 3 no matter what the input was.

Â If we graph this function, you notice all of the points lie on the horizontal line.

Â A lot of students get confused and think that this isn't a function, because all

Â the values were the same. However, notice that we did get this, for

Â each input, there was only one output. It wasn't like I put in a number and got

Â two different outputs. So this actually is a function.

Â The function value that it is, is h of x equals 3.

Â No matter what number I input, the output is always 3, and the graph of this

Â function looks like a horizontal line. Let's try one last function, r of t.

Â Notice that my variable on this case for the input is t.

Â Normally, we use x but there's nothing special about x.

Â We just tend to use that a lot in mathematics.

Â But the variable could be any letter. In this case, our function is r is a

Â function of t. Let's go ahead and try out some values.

Â Victor, you've got an input for me? Let's try 2, r of 2 is pi.

Â Let's try another one, 0. Let's plug in 0, r of 0 is scissors.

Â Let's try one more. Let's put in a chick, r of a chicken is,

Â let's see here what do we got. A bunny [laugh].

Â Let's try another one. Okay.

Â Let's input red, r of red is blue. So, we put in some values to our function,

Â r of t. Victor, up till now, does it look like

Â this is a function? Sure seems like it's a function so far,

Â but to test this, let's try putting in one of the inputs we've already tried.

Â Can I have a chick again? Last time when I put a chicken, I got a

Â bunny. Let's see what happens this time.

Â I'll put my chicken to my function and out comes a sock.

Â One time I put in a chick and I got a bunny, another time, I got a dirty sock.

Â This is an example of something that is not a function.

Â Do you think it's a function now, Victor? No, this isn't a function, because

Â sometimes you put in a chicken, you got a bunny, other times, you got a sock.

Â In the example we just did, the function r of t was looking pretty good to start out

Â with. For each number I put in or each object I

Â put in, I got a unique output. But then, when I try and check again, one

Â time I got bunny and a second time I got sock.

Â So, for this input I got two different outputs.

Â Let's remind ourselves the definition of function was.

Â A function is a correspondence between two sets of elements such that, for each

Â element in the first set, there corresponds one and only one element of

Â the second set. The example we just tried violates this

Â definition, because, sometimes for chick, I got a bunny, sometimes I got a sock.

Â You shouldn't think that these things that aren't functions are very exotic, strange

Â things that you would never encounter. A lot of the things we deal with

Â mathematics aren't functions. For example, circles are not typically

Â functions, because, it's possible to have a circle where you put in one x value and

Â you get two different y values. For example, a circle with radius 1, when

Â you plug in x, you might get y equals 1 or you might get y equals negative 1.

Â So it's not a function. Let's go ahead and talk about what the

Â domain and range of these objects that we've been dealing with, these functions.

Â Back to our function. The domain is a set of things we're

Â allowed to put into the function. What are the allowable inputs to our

Â function? So I input the domain and the things I get

Â out of the function, the outputs are my range.

Â So the domain is the inputs, the range is the output.

Â When looking for a domain, we usually want to think about what is allowed to be put

Â into the function. The usual things we look for is, remember,

Â you're never allowed to divide by zero. So, we look for places where there's

Â divisions by zero. We also know that you're not allowed to

Â take the square root of a negative, so we look for places where you might be taking

Â the square root of a negative. Those are two common examples of things we

Â look for when finding the domain of a function.

Â The range, we usually look to see what are the allowable outputs for the given

Â domain. In this unit on functions and graphs, you

Â will learn to, identify functions. That is, see whether it satisfies the rule

Â that for each input, there is one and only one output.

Â You'll learn to evaluate functions. That's what we're doing with our box, but

Â you'll be doing that symbolically in the graphs.

Â For a given input, figuring out what the output value is.

Â You'll be learning to find the domain and range of functions, what is the set of

Â allowable inputs and what are the corresponding outputs.

Â You'll also graph and find equations for linear functions.

Â Linear functions are special type of function and they're super important in

Â mathematics and come up quite a lot. Linear equations are equations of the

Â form, f of x equals mx plus b. The graph of these functions are straight

Â lines with various slopes, m, m represents the slope or how steep this linear

Â function is. The domain in range for the linear

Â functions are unique and that they're both all real numbers.

Â You can put any real number in, you can get any real number out.

Â You'll also learn how to find the line, the slope of the line, and the x and y

Â intercepts of a given line in order to help you graph it.

Â Other things you'll learn in this unit are to perform transformations of graphs.

Â Transformations of graphs basically mean you'll start with a standard graph that

Â you know, for example, the parabola graph is a common one you might already know.

Â And you learn what happens when you do slight changes to it, for example, adding

Â 2 to the parabola, x squared, moves the graph up two units.

Â Subtracting 3, x minus 3, and then squaring it causes the graph to move three

Â units to the side. We could also do things like stretch or

Â expand our graph. For example, we have the graph here of 5x

Â squared, which causes a vertical stretching of my graph.

Â Another thing I could do is multiply by a negative number, if I look at f of x

Â equals negative x squared, it causes my graph to reflect or give the mirror image

Â over the x-axis. You'll be learning about these sorts of

Â transformations in the coming unit. When talking about combinations of

Â functions, you'll learn to do things like f plus g, f minus g, f times g, f divided

Â g, and something called composition. F composed with g, means, we do g of x

Â first and then we do f, the resulting output from g.

Â It's basically like having two of my machine function boxes in a row, where one

Â feeds into the other and then you get an output after you've done both systems

Â together. The last thing you'll be learning about is

Â determining if a function is invertible and finding its inverse.

Â Inverse functions basically lets you go forwards and backwards to recover the

Â original inputs. There's a lot of applications of

Â functions. Basically, the applications or functions

Â are everything around you. For example, if you're running, you might

Â have a function of the amount of calories you burn as a function of the distance

Â you've run. When you put your money in a bank, in an

Â investment, you usually have a function the amount of money you make as a function

Â of the interest rate or the amount of money you make as a function of the time

Â you left that investment in the bank. Well, have fun exploring functions.

Â Thank you and I'll see you next time. [music].

Â