WEBVTT

00:00:00.000 --> 00:00:02.460
In this video, I want
to do a few examples

00:00:02.460 --> 00:00:03.800
dealing with functions.

00:00:03.800 --> 00:00:06.570
Functions tend to be something
that a lot of students find

00:00:06.570 --> 00:00:09.230
difficult, but I think if you
really get what we're talking

00:00:09.230 --> 00:00:11.070
about, you'll see that it's
actually a pretty

00:00:11.070 --> 00:00:12.240
straightforward idea.

00:00:12.240 --> 00:00:13.710
And you sometimes wonder,
well what was all

00:00:13.710 --> 00:00:14.880
of the hubbub about?

00:00:14.880 --> 00:00:16.720
All a function is, is an

00:00:16.720 --> 00:00:19.830
association between two variables.

00:00:19.830 --> 00:00:25.540
So if we say that y is equal to
a function of x, all that

00:00:25.540 --> 00:00:28.260
means is, you give me an x.

00:00:28.260 --> 00:00:31.660
You can imagine this function
is kind of eating up this x.

00:00:31.660 --> 00:00:34.190
You pop an x into
this function.

00:00:34.190 --> 00:00:36.480
This function is just
a set of rules.

00:00:36.480 --> 00:00:39.150
It's going to say, oh,
with that x, I

00:00:39.150 --> 00:00:41.230
associate some value y.

00:00:41.230 --> 00:00:42.945
You can imagine it is
some type of a box.

00:00:45.900 --> 00:00:47.990
That is a function.

00:00:47.990 --> 00:00:53.830
When I give it some number
x, it'll give me some

00:00:53.830 --> 00:00:56.990
other number y.

00:00:56.990 --> 00:00:58.160
This might seem a
little abstract.

00:00:58.160 --> 00:00:59.360
What are these x's and y's?

00:00:59.360 --> 00:01:02.830
Maybe I have a function-- let
me make it like this.

00:01:02.830 --> 00:01:04.190
Let's say I have a function
definition

00:01:04.190 --> 00:01:05.720
that looks like this.

00:01:05.720 --> 00:01:11.770
For any x you give me, I'm going
to produce 1 if x is

00:01:11.770 --> 00:01:14.440
equal to-- I don't know-- 0.

00:01:14.440 --> 00:01:18.730
I'm going to produce 2
if x is equal to 1.

00:01:18.730 --> 00:01:21.320
And I'm going to produce
3 otherwise.

00:01:24.790 --> 00:01:28.720
So now we've defined what's
going on inside of the box.

00:01:28.720 --> 00:01:31.630
So let's draw the
box around it.

00:01:31.630 --> 00:01:33.650
This is our box.

00:01:33.650 --> 00:01:35.940
This is just an arbitrary
function definition, but

00:01:35.940 --> 00:01:37.760
hopefully it'll help you
understand what's actually

00:01:37.760 --> 00:01:40.070
going on with a function.

00:01:40.070 --> 00:01:47.500
So now if I make x is equal to--
if I pick x is equal to

00:01:47.500 --> 00:01:52.480
7, now what is f of x going
to be equal to?

00:01:52.480 --> 00:01:56.400
What is f of 7 going
to be equal to?

00:01:56.400 --> 00:01:58.020
So I take 7 into the box.

00:01:58.020 --> 00:01:59.700
You could view it as some
type of a computer.

00:01:59.700 --> 00:02:02.770
The computer looks at that x and
then looks at its rules.

00:02:02.770 --> 00:02:04.060
It says, OK. x is 7.

00:02:04.060 --> 00:02:06.270
Well x isn't 0. x isn't 1.

00:02:06.270 --> 00:02:08.229
I go to the otherwise
situation.

00:02:08.229 --> 00:02:10.100
So I'm going to pop out a 3.

00:02:10.100 --> 00:02:12.040
So f of 7 is equal to 3.

00:02:12.040 --> 00:02:15.320
So we write f of 7
is equal to 3.

00:02:15.320 --> 00:02:18.760
Where f is the name of this
function, this rule system, or

00:02:18.760 --> 00:02:21.310
this association, this
mapping, whatever you

00:02:21.310 --> 00:02:22.190
want to call it.

00:02:22.190 --> 00:02:24.350
When you give it a 7,
it'll produce a 3.

00:02:24.350 --> 00:02:27.460
When you give f a 7,
it'll produce a 3.

00:02:27.460 --> 00:02:31.240
What is f of 2?

00:02:31.240 --> 00:02:34.690
Well, that means instead of x
is equal to 7, I'm going to

00:02:34.690 --> 00:02:36.420
give it an x equal 2.

00:02:36.420 --> 00:02:38.550
Then the little computer inside
the function is going

00:02:38.550 --> 00:02:42.550
to say, OK, let's see,
when x is equal to 2.

00:02:42.550 --> 00:02:44.410
No, I'm still in the otherwise
situation.

00:02:44.410 --> 00:02:45.910
x isn't 0 or 1.

00:02:45.910 --> 00:02:50.800
So once again f of
x is equal to 3.

00:02:53.470 --> 00:02:56.970
So, this is f of 2 is
also equal to 3.

00:02:56.970 --> 00:03:03.200
Now what happens if x
is now equal to 1?

00:03:03.200 --> 00:03:05.100
Well then it's just going
to turn over this.

00:03:05.100 --> 00:03:07.990
So f of 1.

00:03:07.990 --> 00:03:10.080
It's going to look at its
rules right here.

00:03:10.080 --> 00:03:11.620
Oh look, x is equal to 1.

00:03:11.620 --> 00:03:13.350
I can use my rule right here.

00:03:13.350 --> 00:03:15.520
So when x is equal to
1, I spit out a 2.

00:03:15.520 --> 00:03:18.750
So f of 1 is going
to be equal to 2.

00:03:18.750 --> 00:03:22.290
I spit out f of 1, which is
equal to 2 in that situation.

00:03:22.290 --> 00:03:24.420
That's all a function is.

00:03:24.420 --> 00:03:29.120
Now, with that in mind, let's
do some of these example

00:03:29.120 --> 00:03:31.620
problems. They tell us for
each of the following

00:03:31.620 --> 00:03:35.010
functions, evaluate these
different functions-- these

00:03:35.010 --> 00:03:37.570
are the different boxes they've
created-- at these

00:03:37.570 --> 00:03:39.070
different points.

00:03:39.070 --> 00:03:42.800
Let's do part a first. They're
defining the box.

00:03:42.800 --> 00:03:47.880
f of x is equal to negative
2x plus 3.

00:03:47.880 --> 00:03:51.790
They want to know what happens
when f is equal to negative 3.

00:03:51.790 --> 00:03:54.300
Well f is equal to negative 3,
this is telling me what do I

00:03:54.300 --> 00:03:55.430
do with the x?

00:03:55.430 --> 00:03:57.110
What do I produce?

00:03:57.110 --> 00:04:00.060
Wherever I see an x, I replace
it with the negative 3.

00:04:00.060 --> 00:04:02.060
So it's going to be equal
to negative 2.

00:04:02.060 --> 00:04:04.780
Let me do it this way, so you
see exactly what I'm doing.

00:04:04.780 --> 00:04:06.520
That negative 3, I'll do
it in that bold color.

00:04:06.520 --> 00:04:13.130
It's negative 2 times
negative 3 plus 3.

00:04:13.130 --> 00:04:16.149
Notice wherever there was an
x, I put the negative 3.

00:04:16.149 --> 00:04:19.250
So I know what the black box
is going to produce.

00:04:19.250 --> 00:04:21.600
This is going to be equal to
negative 2 times negative 3 is

00:04:21.600 --> 00:04:25.640
6 plus 3, which is equal to 9.

00:04:25.640 --> 00:04:29.470
So f of negative 3
is equal to 9.

00:04:29.470 --> 00:04:32.130
What about f of 7?

00:04:32.130 --> 00:04:36.340
I'll do the same thing one more
time. f of-- I'll do 7 in

00:04:36.340 --> 00:04:43.120
yellow-- f of 7 is going to
be equal to negative 2

00:04:43.120 --> 00:04:47.650
times 7 plus 3.

00:04:50.480 --> 00:04:55.140
So this is equal to negative 14
plus 3, which is equal to

00:04:55.140 --> 00:04:57.260
negative 11.

00:04:57.260 --> 00:05:03.940
You put in-- let me make it very
clear-- you put in a 7

00:05:03.940 --> 00:05:11.060
into our function f here and it
will pop out a negative 11.

00:05:11.060 --> 00:05:13.310
That's what this just
told us right there.

00:05:13.310 --> 00:05:14.760
This is the rule.

00:05:14.760 --> 00:05:18.470
This is completely analogous
to what I did up here.

00:05:18.470 --> 00:05:20.980
This is the rule of
our function.

00:05:20.980 --> 00:05:24.430
Let's do the next two.

00:05:24.430 --> 00:05:25.200
I won't do part b.

00:05:25.200 --> 00:05:26.330
You can do part b for fun.

00:05:26.330 --> 00:05:29.650
I'll do part c after that, just
for the sake of time.

00:05:29.650 --> 00:05:32.540
Now we are at f of 0.

00:05:32.540 --> 00:05:33.810
Here I'll just do
it in one color.

00:05:33.810 --> 00:05:35.300
I think you're getting
the idea. f of 0.

00:05:35.300 --> 00:05:37.500
Wherever we see an
x, we put a 0.

00:05:37.500 --> 00:05:40.005
So negative 2 times 0 plus 3.

00:05:43.100 --> 00:05:44.345
Well, that's just
going to be a 0.

00:05:44.345 --> 00:05:47.300
So f of 0 is 3.

00:05:47.300 --> 00:05:49.000
Then one last one. f of z.

00:05:49.000 --> 00:05:51.720
They want to keep it
abstract for us.

00:05:51.720 --> 00:05:52.780
Here I'll color code it.

00:05:52.780 --> 00:05:55.800
So f of z.

00:05:55.800 --> 00:05:59.150
Let me make the z in
a different color.

00:05:59.150 --> 00:06:00.900
f of z.

00:06:00.900 --> 00:06:06.210
Everywhere that we saw
an x, we will now

00:06:06.210 --> 00:06:07.750
replace it with a z.

00:06:07.750 --> 00:06:09.240
Negative 2.

00:06:09.240 --> 00:06:12.040
Instead of an x, we're going
to put a z there.

00:06:12.040 --> 00:06:13.860
We're going to put an
orange z there.

00:06:13.860 --> 00:06:19.760
Negative 2 times z plus 3.

00:06:19.760 --> 00:06:24.330
And that's our answer. f of
z is negative 2z plus 3.

00:06:24.330 --> 00:06:28.110
If you imagine our box,
the function f.

00:06:28.110 --> 00:06:38.130
You put in a z, you are going to
get out a negative 2 times

00:06:38.130 --> 00:06:43.480
whatever that z is plus 3.

00:06:43.480 --> 00:06:44.520
That's all this is saying.

00:06:44.520 --> 00:06:47.830
It's a little bit more abstract,
but same exact idea.

00:06:47.830 --> 00:06:52.030
Now let's just do part c here.

00:06:52.030 --> 00:06:53.330
Let me clear this actually.

00:06:53.330 --> 00:06:55.820
I'm running out of
real estate.

00:06:55.820 --> 00:06:59.102
Let me clear all of
this business.

00:06:59.102 --> 00:07:02.910
Let me clear all of
this business.

00:07:02.910 --> 00:07:03.810
We can do part c.

00:07:03.810 --> 00:07:05.370
I'm skipping part b.

00:07:05.370 --> 00:07:07.710
You can work on that part.

00:07:07.710 --> 00:07:10.830
Part b.

00:07:10.830 --> 00:07:13.430
They tell us-- this is our
function definition.

00:07:13.430 --> 00:07:16.680
Sorry, I said I was
doing part c.

00:07:16.680 --> 00:07:18.610
This is our function
definition.

00:07:18.610 --> 00:07:26.300
f of x is equal to 5 times
2 minus x over 11.

00:07:26.300 --> 00:07:29.440
So let's apply these different
values of x, these different

00:07:29.440 --> 00:07:32.620
inputs into our function.

00:07:32.620 --> 00:07:39.900
So f of negative 3 is equal to
5 times 2 minus-- wherever we

00:07:39.900 --> 00:07:42.250
see an x, we put a negative 3.

00:07:42.250 --> 00:07:45.620
2 minus negative 3 over 11.

00:07:45.620 --> 00:07:48.700
This is equal to 2 plus 3.

00:07:48.700 --> 00:07:50.870
This is equal to 5.

00:07:50.870 --> 00:07:53.260
So you get 5 times 5 over 11.

00:07:53.260 --> 00:07:57.120
That's equal to 25/11.

00:07:57.120 --> 00:07:57.850
Let's do this one.

00:07:57.850 --> 00:07:59.990
F of 7.

00:07:59.990 --> 00:08:06.680
For this second function right
here, f of 7 is equal to 5

00:08:06.680 --> 00:08:11.160
times 2 minus-- now
for x we have a 7.

00:08:11.160 --> 00:08:14.360
2 minus 7 over 11.

00:08:14.360 --> 00:08:15.540
So what is this going
to be equal to?

00:08:15.540 --> 00:08:18.250
2 minus 7 is negative 5.

00:08:18.250 --> 00:08:23.780
5 times negative 5 is
negative 25/11.

00:08:23.780 --> 00:08:27.410
Then finally, well we have
two more. f of 0.

00:08:27.410 --> 00:08:35.000
That's equal to 5 times 2 minus
0 So this is just 2.

00:08:35.000 --> 00:08:36.130
5 times 2 is 10.

00:08:36.130 --> 00:08:38.850
So this is equal to 10/11.

00:08:38.850 --> 00:08:39.840
One more.

00:08:39.840 --> 00:08:42.059
f of z.

00:08:42.059 --> 00:08:43.299
Well every time we saw
an x, we're going to

00:08:43.299 --> 00:08:44.490
replace it with a z.

00:08:44.490 --> 00:08:49.960
It's equal to 5 times
2 minus z over 11.

00:08:49.960 --> 00:08:50.630
And that's our answer.

00:08:50.630 --> 00:08:51.910
We could distribute the 5.

00:08:51.910 --> 00:08:57.210
You could say this is the same
thing as 10 minus 5z over 11.

00:08:57.210 --> 00:09:00.260
We could even write it in
slope-intercept form.

00:09:00.260 --> 00:09:06.000
This is the same thing as
minus 5/11 z plus 10/11.

00:09:06.000 --> 00:09:06.990
These are all equivalent.

00:09:06.990 --> 00:09:10.430
But that is what f
of z is equal to.

00:09:10.430 --> 00:09:11.590
Now.

00:09:11.590 --> 00:09:15.510
A function, we said, if you give
me any x value, I will

00:09:15.510 --> 00:09:16.470
give you an output.

00:09:16.470 --> 00:09:19.120
I will give you an f of x.

00:09:19.120 --> 00:09:23.040
So if this is our function,
you give me an x, it will

00:09:23.040 --> 00:09:26.550
produce an f of x.

00:09:26.550 --> 00:09:29.680
It can only produce 1
f of x for any x.

00:09:29.680 --> 00:09:32.840
You can't have a function that
could produce two possible

00:09:32.840 --> 00:09:34.700
values for an x.

00:09:34.700 --> 00:09:37.540
So you can't have a function--
this would be an invalid

00:09:37.540 --> 00:09:42.790
function definition-- f
of x is equal to 3 if

00:09:42.790 --> 00:09:45.230
x is equal to 0.

00:09:45.230 --> 00:09:49.240
Or it could be equal to
4 if x is equal to 0.

00:09:49.240 --> 00:09:53.170
Because in this situation, we
don't know what f of 0 is.

00:09:53.170 --> 00:09:54.090
What it's going to be equal?

00:09:54.090 --> 00:09:56.330
It says if x is equal to 0, it
should be 3 or it could be--

00:09:56.330 --> 00:09:57.310
we don't know.

00:09:57.310 --> 00:09:57.830
We don't know.

00:09:57.830 --> 00:09:58.190
We don't know.

00:09:58.190 --> 00:10:01.550
This is not a function
even though it might

00:10:01.550 --> 00:10:02.800
have looked like one.

00:10:07.700 --> 00:10:12.250
So you can't have two f of
x values for one x value.

00:10:12.250 --> 00:10:16.020
So let's see which of these
graphs are functions.

00:10:16.020 --> 00:10:18.390
To figure that out, you could
say, look at any x value

00:10:18.390 --> 00:10:21.850
here-- pick any x value-- I have
exactly one f of x value.

00:10:21.850 --> 00:10:25.090
This is y is equal to
f of x right here.

00:10:25.090 --> 00:10:28.950
I have exactly only one--
at that x, that

00:10:28.950 --> 00:10:30.550
is my y value here.

00:10:30.550 --> 00:10:32.970
So you could have a vertical
line test, which says at any

00:10:32.970 --> 00:10:35.720
point if you draw a vertical
line-- notice a vertical line

00:10:35.720 --> 00:10:37.570
is for a certain x value.

00:10:37.570 --> 00:10:41.920
That shows that I only have
one y value at that point.

00:10:41.920 --> 00:10:43.630
So this is a valid function.

00:10:43.630 --> 00:10:46.240
Any time you draw a vertical
line, it will only intersect

00:10:46.240 --> 00:10:47.610
the graph once.

00:10:47.610 --> 00:10:50.410
So this is a valid function.

00:10:50.410 --> 00:10:52.220
Now what about this
one right here?

00:10:52.220 --> 00:10:53.960
I could draw a vertical
line, let's say, at

00:10:53.960 --> 00:10:55.230
that point right there.

00:10:55.230 --> 00:10:58.650
For that x, this relation
seems to have two

00:10:58.650 --> 00:11:00.860
possible f of x's.

00:11:00.860 --> 00:11:04.550
f of x could be that value or
f of x could be that value.

00:11:04.550 --> 00:11:05.270
Right?

00:11:05.270 --> 00:11:07.520
We're intersecting
the graph twice.

00:11:07.520 --> 00:11:08.840
So this is not a function.

00:11:08.840 --> 00:11:11.150
We're doing exactly what
I described here.

00:11:11.150 --> 00:11:15.090
For a certain x, we're
describing two possible y's

00:11:15.090 --> 00:11:16.800
that could be equal to f of x.

00:11:16.800 --> 00:11:19.220
So this is not a function.

00:11:19.220 --> 00:11:20.830
Over here, same thing.

00:11:20.830 --> 00:11:22.310
You draw a vertical
line right there.

00:11:22.310 --> 00:11:24.540
You're intersecting
the graph twice.

00:11:24.540 --> 00:11:26.000
This is not a function.

00:11:26.000 --> 00:11:30.590
You're defining two possible
y values for 1 x value.

00:11:30.590 --> 00:11:31.490
Let's go to this function.

00:11:31.490 --> 00:11:33.160
It's kind of a weird
looking function.

00:11:33.160 --> 00:11:34.750
Looks like a reversed
check mark.

00:11:34.750 --> 00:11:37.020
But any time you draw a vertical
line, you're only

00:11:37.020 --> 00:11:38.720
intersecting it once.

00:11:38.720 --> 00:11:40.420
So this is a valid function.

00:11:40.420 --> 00:11:43.470
For every x, you only have
one y associated.

00:11:43.470 --> 00:11:46.450
Or only one f of x associated
with it.

00:11:46.450 --> 00:11:48.960
Anyway, hopefully you
found that useful.