Functions Part 1

Uploaded by videosbyjulieharland on 25.05.2010

>> In this video I will be introducing the idea
of what a relation and a function is.
Both of these could be represented in different ways.
For instance we could have a table of values, a graph,
a formula; this is the most common way you'll see.
Words where it's described,
a set of ordered pairs and a picture.
So let's see what I mean by that.
Alright so a relation is a pairing of 2 things.
There is an input and out comes and output
and so one way you might do it lets say I've got this picture
and I've got some numbers to make it easy the numbers 1, 2 ,
and 4 and then I have this output and let's say
over here I have a 3, 4, and a 5
so what I could do is draw a picture
to say what happens each input might have you know more
than one output or just one output.
So here's an example of a picture of a relation.
Now what its saying is the input is over here on the left
and the output is over here on the right.
Now if I was gonna write that as a set of ordered pairs,
notice I have if I put in a 1 I get a 3, also if I put
in a 1 I get a 4, right ok so those take care of those 2 lines
and then I've got 2 also going to 4.
So here's another ordered pair 2, 4.
I have this other ordered pair 3,
5 and say I have an ordered pair 4, 5.
So notice I have 4 lines here so there should be 4 ordered pairs
and this is another way of showing this relation
and we could write that as a set so we could use braces.
We could also do that in a little table,
remember I said you could do it in a little table
and we could say well let's say the input we'll call like X
if we want and the output we could call Y,
you could write the word input and output.
So if the input is 1 the output was 3.
Also when I had an input of 1 I was able to get an output of 4.
When I imputed 2 I was able to get an output of 4.
When I imputed 3 I got an output of 5
and when I input 4 I also got an output of 5
and then I said you could describe this in words.
You would say 1 goes to 3 you could just say
that in words 1 goes to 3, 1 also goes to 4, 2 goes to 4,
3 goes to 5 and 4 goes to 5, I'm not gonna write that all out
but that's how you could describe it in words.
This one doesn't have a particular formula
but we'll be getting into some
of the more we could also write this as a formula
and let's see the other one we did as a graph
so let's say we just did a little graph,
usually what we do is we make the input X and the output Y
and for my inputs I've got the numbers 1, 2 ,
and 4 and for my output I have 3, 4,
and 5 so let's see I'm gonna make the lengths
on the Y axis different than that of the X axis that here's 5
and here's 5 so if I wanted
to draw what those ordered pair are here's 1,
3 and I also have 1, 5 I mean I'm sorry 1, 4 and I've got 2,
4 and I have 3, 5 and I have 4, 5.
So there's like 4 ways
that I could show this particular relation.
We call all of the inputs the domain and we call all
of the outputs the range.
So let's see from this particular picture
if I was going to list what was called the domain that's the set
of inputs ok.
Well what numbers do I have?
I have 1, 2, 3 and 4, that's my domain.
Notice I don't write 1 twice
because it's just a set a numbers
and there's only 4 numbers it's just that 1 is listed more
than once and then for my range that's the set
of outputs what do I have?
All I have is the 3, 4 and 5.
It's easy to see in the picture right.
I've got 1, 2, 3 and 4 for my inputs;
my outputs are 3, 4 and 5.
Alright so that's a lot of vocabulary just to get started
but basically the idea is this relation has some inputs
and outputs.
You might have noticed that if the relation is written as a set
of ordered pairs, or if you're looking at the graph
of the relation the domain is simply the set of X values
and the range is the set of Y values
but you might not be looking at a relation that is written
with numbers in it at all.
So let's look at a different kind of a relation.
How about this relation?
Here's a relation that shows the husbands names of 4 sisters:
Amy, Beth, Cindy and Denise.
Alright to make it simple let's see if I just draw a picture
and I'll use the first letter of their name so I don't have
to write it out and I'm gonna write over here
who they married right so actually Amy
and Beth both marry John then I would draw a picture to say both
of them marry someone named John and let's say
that Cindy married Mark
so I would draw a picture arrow Cindy married Mark
and let's say Denise married Steve and that's his first name.
So if we're just doing the first names that's
for instance a picture of what that relation would look like.
Now you could also write them as ordered pairs, Amy John,
Beth John, Cindy Mark and Denise Steve.
You could write them as an ordered pair it's just
that they don't have any numbers in them.
You should be able to figure out what the domain and range is.
The domain would be Amy, Beth, Cindy and Denise right?
That's the input and the range would be just John,
Mark and Steve.
I could write those out
but hopefully you're getting the idea.
Here's a picture of a relation
and the relation is the ordered pairs that you see.
So see if you can state the relation as a set
of ordered pairs and then state the domain and state the range
and you want to use set notation so use braces.
So put the video on pause and try that.
Alright so let's do it.
We're gonna state the relation as a set of ordered pairs.
I've got 5 ordered pairs
so I'm just gonna list all the ordered pairs
and it doesn't matter the order you write the ordered pairs
in so I notice I have this one ordered pair right here that's
1, 1 and I also have 1 up here that's 1, 4.
Put this one right here it's over 2 and of 1.
Got the 1 right on the X axis here, there's the X
of Y axis right that's 3, 0 and I have one
over here 4 negative 1.
So that's a different way of writing the relation.
If I want to state the domain it's a set of X values
or the inputs or the X values here.
So I've got 1, 2, 3 and 4
and for the range what I have I have a 1, a 4, a 1 again,
0 and negative 1 and it doesn't matter the order you put them
in I'm gonna put them from lowest to smallest.
I've got -1, 0, 1 and 4.
You're basically looking at the Y values over here
so you could see there was a Y value over here, here, here,
here you see that's down at -1, 0,
1 and 5 if you're checking out the Y values.
Alright we're gonna get to just very basic idea of a function.
In a function each input always has exactly 1 output.
You can think of there being
like a little function machine here so a function machine
so if I put something in if I put my input in there it's
like it does some stuff in here ok there's something going
on in here and then out pops an output
and exactly one output comes out of there it's not
like the next time I put
in the exact same thing something different will come
out so on the next video we're going
to concentrate on functions.