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

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

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

functions. Functions come up all the time in the real

world. Let's start by talking about the

definition of function. A function is a correspondence between two

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

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

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

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

function. We usually denote functions by y equals f

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

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

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

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

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

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

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

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

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

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

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

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

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

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

the value of 8. F of 3 equals 8.

Let's try another one. How about 6?

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

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

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

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

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

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

we put panda into the function and out comes tiger.

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

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

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

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

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

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

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

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

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

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

Victor? Let's put in 1.

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

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

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

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

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

We get a, another 7. One more.

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

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

You noticed it seems to have a pretty strong pattern.

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

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

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

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

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

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

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

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

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

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

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

this is probably the function g of x equals x.

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

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

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

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

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

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

H of 1 is 3. Let's try another one.

2, h of 2 is. 3 again.

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

Let's see, 3. H of 3 is 3.

Let's plug in 7 now. H of 7 is 3.

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

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

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

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

Does this contradict the definition of function?

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

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

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

an output of 3 no matter what the input was.

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

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

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

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

two different outputs. So this actually is a function.

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

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

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

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

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

We just tend to use that a lot in mathematics.

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

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

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

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

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

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

Let's try another one. Okay.

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

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

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

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

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

bunny. Let's see what happens this time.

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

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

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

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

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

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

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

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

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

So, for this input I got two different outputs.

Let's remind ourselves the definition of function was.

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

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

the second set. The example we just tried violates this

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

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

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

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

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

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

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

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

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

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

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

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

out of the function, the outputs are my range.

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

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

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

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

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

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

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

look for when finding the domain of a function.

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

domain. In this unit on functions and graphs, you

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

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

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

you'll be doing that symbolically in the graphs.

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

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

allowable inputs and what are the corresponding outputs.

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

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

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

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

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

function is. The domain in range for the linear

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

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

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

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

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

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

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

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

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

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

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

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

squared, which causes a vertical stretching of my graph.

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

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

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

transformations in the coming unit. When talking about combinations of

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

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

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

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

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

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

determining if a function is invertible and finding its inverse.

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

original inputs. There's a lot of applications of

functions. Basically, the applications or functions

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

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

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

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

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

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

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