[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:02.46,Default,,0000,0000,0000,,In this video, I want\Nto do a few examples
Dialogue: 0,0:00:02.46,0:00:03.80,Default,,0000,0000,0000,,dealing with functions.
Dialogue: 0,0:00:03.80,0:00:06.57,Default,,0000,0000,0000,,Functions tend to be something\Nthat a lot of students find
Dialogue: 0,0:00:06.57,0:00:09.23,Default,,0000,0000,0000,,difficult, but I think if you\Nreally get what we're talking
Dialogue: 0,0:00:09.23,0:00:11.07,Default,,0000,0000,0000,,about, you'll see that it's\Nactually a pretty
Dialogue: 0,0:00:11.07,0:00:12.24,Default,,0000,0000,0000,,straightforward idea.
Dialogue: 0,0:00:12.24,0:00:13.71,Default,,0000,0000,0000,,And you sometimes wonder,\Nwell what was all
Dialogue: 0,0:00:13.71,0:00:14.88,Default,,0000,0000,0000,,of the hubbub about?
Dialogue: 0,0:00:14.88,0:00:16.72,Default,,0000,0000,0000,,All a function is, is an
Dialogue: 0,0:00:16.72,0:00:19.83,Default,,0000,0000,0000,,association between two variables.
Dialogue: 0,0:00:19.83,0:00:25.54,Default,,0000,0000,0000,,So if we say that y is equal to\Na function of x, all that
Dialogue: 0,0:00:25.54,0:00:28.26,Default,,0000,0000,0000,,means is, you give me an x.
Dialogue: 0,0:00:28.26,0:00:31.66,Default,,0000,0000,0000,,You can imagine this function\Nis kind of eating up this x.
Dialogue: 0,0:00:31.66,0:00:34.19,Default,,0000,0000,0000,,You pop an x into\Nthis function.
Dialogue: 0,0:00:34.19,0:00:36.48,Default,,0000,0000,0000,,This function is just\Na set of rules.
Dialogue: 0,0:00:36.48,0:00:39.15,Default,,0000,0000,0000,,It's going to say, oh,\Nwith that x, I
Dialogue: 0,0:00:39.15,0:00:41.23,Default,,0000,0000,0000,,associate some value y.
Dialogue: 0,0:00:41.23,0:00:42.94,Default,,0000,0000,0000,,You can imagine it is\Nsome type of a box.
Dialogue: 0,0:00:45.90,0:00:47.99,Default,,0000,0000,0000,,That is a function.
Dialogue: 0,0:00:47.99,0:00:53.83,Default,,0000,0000,0000,,When I give it some number\Nx, it'll give me some
Dialogue: 0,0:00:53.83,0:00:56.99,Default,,0000,0000,0000,,other number y.
Dialogue: 0,0:00:56.99,0:00:58.16,Default,,0000,0000,0000,,This might seem a\Nlittle abstract.
Dialogue: 0,0:00:58.16,0:00:59.36,Default,,0000,0000,0000,,What are these x's and y's?
Dialogue: 0,0:00:59.36,0:01:02.83,Default,,0000,0000,0000,,Maybe I have a function-- let\Nme make it like this.
Dialogue: 0,0:01:02.83,0:01:04.19,Default,,0000,0000,0000,,Let's say I have a function\Ndefinition
Dialogue: 0,0:01:04.19,0:01:05.72,Default,,0000,0000,0000,,that looks like this.
Dialogue: 0,0:01:05.72,0:01:11.77,Default,,0000,0000,0000,,For any x you give me, I'm going\Nto produce 1 if x is
Dialogue: 0,0:01:11.77,0:01:14.44,Default,,0000,0000,0000,,equal to-- I don't know-- 0.
Dialogue: 0,0:01:14.44,0:01:18.73,Default,,0000,0000,0000,,I'm going to produce 2\Nif x is equal to 1.
Dialogue: 0,0:01:18.73,0:01:21.32,Default,,0000,0000,0000,,And I'm going to produce\N3 otherwise.
Dialogue: 0,0:01:24.79,0:01:28.72,Default,,0000,0000,0000,,So now we've defined what's\Ngoing on inside of the box.
Dialogue: 0,0:01:28.72,0:01:31.63,Default,,0000,0000,0000,,So let's draw the\Nbox around it.
Dialogue: 0,0:01:31.63,0:01:33.65,Default,,0000,0000,0000,,This is our box.
Dialogue: 0,0:01:33.65,0:01:35.94,Default,,0000,0000,0000,,This is just an arbitrary\Nfunction definition, but
Dialogue: 0,0:01:35.94,0:01:37.76,Default,,0000,0000,0000,,hopefully it'll help you\Nunderstand what's actually
Dialogue: 0,0:01:37.76,0:01:40.07,Default,,0000,0000,0000,,going on with a function.
Dialogue: 0,0:01:40.07,0:01:47.50,Default,,0000,0000,0000,,So now if I make x is equal to--\Nif I pick x is equal to
Dialogue: 0,0:01:47.50,0:01:52.48,Default,,0000,0000,0000,,7, now what is f of x going\Nto be equal to?
Dialogue: 0,0:01:52.48,0:01:56.40,Default,,0000,0000,0000,,What is f of 7 going\Nto be equal to?
Dialogue: 0,0:01:56.40,0:01:58.02,Default,,0000,0000,0000,,So I take 7 into the box.
Dialogue: 0,0:01:58.02,0:01:59.70,Default,,0000,0000,0000,,You could view it as some\Ntype of a computer.
Dialogue: 0,0:01:59.70,0:02:02.77,Default,,0000,0000,0000,,The computer looks at that x and\Nthen looks at its rules.
Dialogue: 0,0:02:02.77,0:02:04.06,Default,,0000,0000,0000,,It says, OK. x is 7.
Dialogue: 0,0:02:04.06,0:02:06.27,Default,,0000,0000,0000,,Well x isn't 0. x isn't 1.
Dialogue: 0,0:02:06.27,0:02:08.23,Default,,0000,0000,0000,,I go to the otherwise\Nsituation.
Dialogue: 0,0:02:08.23,0:02:10.10,Default,,0000,0000,0000,,So I'm going to pop out a 3.
Dialogue: 0,0:02:10.10,0:02:12.04,Default,,0000,0000,0000,,So f of 7 is equal to 3.
Dialogue: 0,0:02:12.04,0:02:15.32,Default,,0000,0000,0000,,So we write f of 7\Nis equal to 3.
Dialogue: 0,0:02:15.32,0:02:18.76,Default,,0000,0000,0000,,Where f is the name of this\Nfunction, this rule system, or
Dialogue: 0,0:02:18.76,0:02:21.31,Default,,0000,0000,0000,,this association, this\Nmapping, whatever you
Dialogue: 0,0:02:21.31,0:02:22.19,Default,,0000,0000,0000,,want to call it.
Dialogue: 0,0:02:22.19,0:02:24.35,Default,,0000,0000,0000,,When you give it a 7,\Nit'll produce a 3.
Dialogue: 0,0:02:24.35,0:02:27.46,Default,,0000,0000,0000,,When you give f a 7,\Nit'll produce a 3.
Dialogue: 0,0:02:27.46,0:02:31.24,Default,,0000,0000,0000,,What is f of 2?
Dialogue: 0,0:02:31.24,0:02:34.69,Default,,0000,0000,0000,,Well, that means instead of x\Nis equal to 7, I'm going to
Dialogue: 0,0:02:34.69,0:02:36.42,Default,,0000,0000,0000,,give it an x equal 2.
Dialogue: 0,0:02:36.42,0:02:38.55,Default,,0000,0000,0000,,Then the little computer inside\Nthe function is going
Dialogue: 0,0:02:38.55,0:02:42.55,Default,,0000,0000,0000,,to say, OK, let's see,\Nwhen x is equal to 2.
Dialogue: 0,0:02:42.55,0:02:44.41,Default,,0000,0000,0000,,No, I'm still in the otherwise\Nsituation.
Dialogue: 0,0:02:44.41,0:02:45.91,Default,,0000,0000,0000,,x isn't 0 or 1.
Dialogue: 0,0:02:45.91,0:02:50.80,Default,,0000,0000,0000,,So once again f of\Nx is equal to 3.
Dialogue: 0,0:02:53.47,0:02:56.97,Default,,0000,0000,0000,,So, this is f of 2 is\Nalso equal to 3.
Dialogue: 0,0:02:56.97,0:03:03.20,Default,,0000,0000,0000,,Now what happens if x\Nis now equal to 1?
Dialogue: 0,0:03:03.20,0:03:05.10,Default,,0000,0000,0000,,Well then it's just going\Nto turn over this.
Dialogue: 0,0:03:05.10,0:03:07.99,Default,,0000,0000,0000,,So f of 1.
Dialogue: 0,0:03:07.99,0:03:10.08,Default,,0000,0000,0000,,It's going to look at its\Nrules right here.
Dialogue: 0,0:03:10.08,0:03:11.62,Default,,0000,0000,0000,,Oh look, x is equal to 1.
Dialogue: 0,0:03:11.62,0:03:13.35,Default,,0000,0000,0000,,I can use my rule right here.
Dialogue: 0,0:03:13.35,0:03:15.52,Default,,0000,0000,0000,,So when x is equal to\N1, I spit out a 2.
Dialogue: 0,0:03:15.52,0:03:18.75,Default,,0000,0000,0000,,So f of 1 is going\Nto be equal to 2.
Dialogue: 0,0:03:18.75,0:03:22.29,Default,,0000,0000,0000,,I spit out f of 1, which is\Nequal to 2 in that situation.
Dialogue: 0,0:03:22.29,0:03:24.42,Default,,0000,0000,0000,,That's all a function is.
Dialogue: 0,0:03:24.42,0:03:29.12,Default,,0000,0000,0000,,Now, with that in mind, let's\Ndo some of these example
Dialogue: 0,0:03:29.12,0:03:31.62,Default,,0000,0000,0000,,problems. They tell us for\Neach of the following
Dialogue: 0,0:03:31.62,0:03:35.01,Default,,0000,0000,0000,,functions, evaluate these\Ndifferent functions-- these
Dialogue: 0,0:03:35.01,0:03:37.57,Default,,0000,0000,0000,,are the different boxes they've\Ncreated-- at these
Dialogue: 0,0:03:37.57,0:03:39.07,Default,,0000,0000,0000,,different points.
Dialogue: 0,0:03:39.07,0:03:42.80,Default,,0000,0000,0000,,Let's do part a first. They're\Ndefining the box.
Dialogue: 0,0:03:42.80,0:03:47.88,Default,,0000,0000,0000,,f of x is equal to negative\N2x plus 3.
Dialogue: 0,0:03:47.88,0:03:51.79,Default,,0000,0000,0000,,They want to know what happens\Nwhen f is equal to negative 3.
Dialogue: 0,0:03:51.79,0:03:54.30,Default,,0000,0000,0000,,Well f is equal to negative 3,\Nthis is telling me what do I
Dialogue: 0,0:03:54.30,0:03:55.43,Default,,0000,0000,0000,,do with the x?
Dialogue: 0,0:03:55.43,0:03:57.11,Default,,0000,0000,0000,,What do I produce?
Dialogue: 0,0:03:57.11,0:04:00.06,Default,,0000,0000,0000,,Wherever I see an x, I replace\Nit with the negative 3.
Dialogue: 0,0:04:00.06,0:04:02.06,Default,,0000,0000,0000,,So it's going to be equal\Nto negative 2.
Dialogue: 0,0:04:02.06,0:04:04.78,Default,,0000,0000,0000,,Let me do it this way, so you\Nsee exactly what I'm doing.
Dialogue: 0,0:04:04.78,0:04:06.52,Default,,0000,0000,0000,,That negative 3, I'll do\Nit in that bold color.
Dialogue: 0,0:04:06.52,0:04:13.13,Default,,0000,0000,0000,,It's negative 2 times\Nnegative 3 plus 3.
Dialogue: 0,0:04:13.13,0:04:16.15,Default,,0000,0000,0000,,Notice wherever there was an\Nx, I put the negative 3.
Dialogue: 0,0:04:16.15,0:04:19.25,Default,,0000,0000,0000,,So I know what the black box\Nis going to produce.
Dialogue: 0,0:04:19.25,0:04:21.60,Default,,0000,0000,0000,,This is going to be equal to\Nnegative 2 times negative 3 is
Dialogue: 0,0:04:21.60,0:04:25.64,Default,,0000,0000,0000,,6 plus 3, which is equal to 9.
Dialogue: 0,0:04:25.64,0:04:29.47,Default,,0000,0000,0000,,So f of negative 3\Nis equal to 9.
Dialogue: 0,0:04:29.47,0:04:32.13,Default,,0000,0000,0000,,What about f of 7?
Dialogue: 0,0:04:32.13,0:04:36.34,Default,,0000,0000,0000,,I'll do the same thing one more\Ntime. f of-- I'll do 7 in
Dialogue: 0,0:04:36.34,0:04:43.12,Default,,0000,0000,0000,,yellow-- f of 7 is going to\Nbe equal to negative 2
Dialogue: 0,0:04:43.12,0:04:47.65,Default,,0000,0000,0000,,times 7 plus 3.
Dialogue: 0,0:04:50.48,0:04:55.14,Default,,0000,0000,0000,,So this is equal to negative 14\Nplus 3, which is equal to
Dialogue: 0,0:04:55.14,0:04:57.26,Default,,0000,0000,0000,,negative 11.
Dialogue: 0,0:04:57.26,0:05:03.94,Default,,0000,0000,0000,,You put in-- let me make it very\Nclear-- you put in a 7
Dialogue: 0,0:05:03.94,0:05:11.06,Default,,0000,0000,0000,,into our function f here and it\Nwill pop out a negative 11.
Dialogue: 0,0:05:11.06,0:05:13.31,Default,,0000,0000,0000,,That's what this just\Ntold us right there.
Dialogue: 0,0:05:13.31,0:05:14.76,Default,,0000,0000,0000,,This is the rule.
Dialogue: 0,0:05:14.76,0:05:18.47,Default,,0000,0000,0000,,This is completely analogous\Nto what I did up here.
Dialogue: 0,0:05:18.47,0:05:20.98,Default,,0000,0000,0000,,This is the rule of\Nour function.
Dialogue: 0,0:05:20.98,0:05:24.43,Default,,0000,0000,0000,,Let's do the next two.
Dialogue: 0,0:05:24.43,0:05:25.20,Default,,0000,0000,0000,,I won't do part b.
Dialogue: 0,0:05:25.20,0:05:26.33,Default,,0000,0000,0000,,You can do part b for fun.
Dialogue: 0,0:05:26.33,0:05:29.65,Default,,0000,0000,0000,,I'll do part c after that, just\Nfor the sake of time.
Dialogue: 0,0:05:29.65,0:05:32.54,Default,,0000,0000,0000,,Now we are at f of 0.
Dialogue: 0,0:05:32.54,0:05:33.81,Default,,0000,0000,0000,,Here I'll just do\Nit in one color.
Dialogue: 0,0:05:33.81,0:05:35.30,Default,,0000,0000,0000,,I think you're getting\Nthe idea. f of 0.
Dialogue: 0,0:05:35.30,0:05:37.50,Default,,0000,0000,0000,,Wherever we see an\Nx, we put a 0.
Dialogue: 0,0:05:37.50,0:05:40.00,Default,,0000,0000,0000,,So negative 2 times 0 plus 3.
Dialogue: 0,0:05:43.10,0:05:44.34,Default,,0000,0000,0000,,Well, that's just\Ngoing to be a 0.
Dialogue: 0,0:05:44.34,0:05:47.30,Default,,0000,0000,0000,,So f of 0 is 3.
Dialogue: 0,0:05:47.30,0:05:49.00,Default,,0000,0000,0000,,Then one last one. f of z.
Dialogue: 0,0:05:49.00,0:05:51.72,Default,,0000,0000,0000,,They want to keep it\Nabstract for us.
Dialogue: 0,0:05:51.72,0:05:52.78,Default,,0000,0000,0000,,Here I'll color code it.
Dialogue: 0,0:05:52.78,0:05:55.80,Default,,0000,0000,0000,,So f of z.
Dialogue: 0,0:05:55.80,0:05:59.15,Default,,0000,0000,0000,,Let me make the z in\Na different color.
Dialogue: 0,0:05:59.15,0:06:00.90,Default,,0000,0000,0000,,f of z.
Dialogue: 0,0:06:00.90,0:06:06.21,Default,,0000,0000,0000,,Everywhere that we saw\Nan x, we will now
Dialogue: 0,0:06:06.21,0:06:07.75,Default,,0000,0000,0000,,replace it with a z.
Dialogue: 0,0:06:07.75,0:06:09.24,Default,,0000,0000,0000,,Negative 2.
Dialogue: 0,0:06:09.24,0:06:12.04,Default,,0000,0000,0000,,Instead of an x, we're going\Nto put a z there.
Dialogue: 0,0:06:12.04,0:06:13.86,Default,,0000,0000,0000,,We're going to put an\Norange z there.
Dialogue: 0,0:06:13.86,0:06:19.76,Default,,0000,0000,0000,,Negative 2 times z plus 3.
Dialogue: 0,0:06:19.76,0:06:24.33,Default,,0000,0000,0000,,And that's our answer. f of\Nz is negative 2z plus 3.
Dialogue: 0,0:06:24.33,0:06:28.11,Default,,0000,0000,0000,,If you imagine our box,\Nthe function f.
Dialogue: 0,0:06:28.11,0:06:38.13,Default,,0000,0000,0000,,You put in a z, you are going to\Nget out a negative 2 times
Dialogue: 0,0:06:38.13,0:06:43.48,Default,,0000,0000,0000,,whatever that z is plus 3.
Dialogue: 0,0:06:43.48,0:06:44.52,Default,,0000,0000,0000,,That's all this is saying.
Dialogue: 0,0:06:44.52,0:06:47.83,Default,,0000,0000,0000,,It's a little bit more abstract,\Nbut same exact idea.
Dialogue: 0,0:06:47.83,0:06:52.03,Default,,0000,0000,0000,,Now let's just do part c here.
Dialogue: 0,0:06:52.03,0:06:53.33,Default,,0000,0000,0000,,Let me clear this actually.
Dialogue: 0,0:06:53.33,0:06:55.82,Default,,0000,0000,0000,,I'm running out of\Nreal estate.
Dialogue: 0,0:06:55.82,0:06:59.10,Default,,0000,0000,0000,,Let me clear all of\Nthis business.
Dialogue: 0,0:06:59.10,0:07:02.91,Default,,0000,0000,0000,,Let me clear all of\Nthis business.
Dialogue: 0,0:07:02.91,0:07:03.81,Default,,0000,0000,0000,,We can do part c.
Dialogue: 0,0:07:03.81,0:07:05.37,Default,,0000,0000,0000,,I'm skipping part b.
Dialogue: 0,0:07:05.37,0:07:07.71,Default,,0000,0000,0000,,You can work on that part.
Dialogue: 0,0:07:07.71,0:07:10.83,Default,,0000,0000,0000,,Part b.
Dialogue: 0,0:07:10.83,0:07:13.43,Default,,0000,0000,0000,,They tell us-- this is our\Nfunction definition.
Dialogue: 0,0:07:13.43,0:07:16.68,Default,,0000,0000,0000,,Sorry, I said I was\Ndoing part c.
Dialogue: 0,0:07:16.68,0:07:18.61,Default,,0000,0000,0000,,This is our function\Ndefinition.
Dialogue: 0,0:07:18.61,0:07:26.30,Default,,0000,0000,0000,,f of x is equal to 5 times\N2 minus x over 11.
Dialogue: 0,0:07:26.30,0:07:29.44,Default,,0000,0000,0000,,So let's apply these different\Nvalues of x, these different
Dialogue: 0,0:07:29.44,0:07:32.62,Default,,0000,0000,0000,,inputs into our function.
Dialogue: 0,0:07:32.62,0:07:39.90,Default,,0000,0000,0000,,So f of negative 3 is equal to\N5 times 2 minus-- wherever we
Dialogue: 0,0:07:39.90,0:07:42.25,Default,,0000,0000,0000,,see an x, we put a negative 3.
Dialogue: 0,0:07:42.25,0:07:45.62,Default,,0000,0000,0000,,2 minus negative 3 over 11.
Dialogue: 0,0:07:45.62,0:07:48.70,Default,,0000,0000,0000,,This is equal to 2 plus 3.
Dialogue: 0,0:07:48.70,0:07:50.87,Default,,0000,0000,0000,,This is equal to 5.
Dialogue: 0,0:07:50.87,0:07:53.26,Default,,0000,0000,0000,,So you get 5 times 5 over 11.
Dialogue: 0,0:07:53.26,0:07:57.12,Default,,0000,0000,0000,,That's equal to 25/11.
Dialogue: 0,0:07:57.12,0:07:57.85,Default,,0000,0000,0000,,Let's do this one.
Dialogue: 0,0:07:57.85,0:07:59.99,Default,,0000,0000,0000,,F of 7.
Dialogue: 0,0:07:59.99,0:08:06.68,Default,,0000,0000,0000,,For this second function right\Nhere, f of 7 is equal to 5
Dialogue: 0,0:08:06.68,0:08:11.16,Default,,0000,0000,0000,,times 2 minus-- now\Nfor x we have a 7.
Dialogue: 0,0:08:11.16,0:08:14.36,Default,,0000,0000,0000,,2 minus 7 over 11.
Dialogue: 0,0:08:14.36,0:08:15.54,Default,,0000,0000,0000,,So what is this going\Nto be equal to?
Dialogue: 0,0:08:15.54,0:08:18.25,Default,,0000,0000,0000,,2 minus 7 is negative 5.
Dialogue: 0,0:08:18.25,0:08:23.78,Default,,0000,0000,0000,,5 times negative 5 is\Nnegative 25/11.
Dialogue: 0,0:08:23.78,0:08:27.41,Default,,0000,0000,0000,,Then finally, well we have\Ntwo more. f of 0.
Dialogue: 0,0:08:27.41,0:08:35.00,Default,,0000,0000,0000,,That's equal to 5 times 2 minus\N0 So this is just 2.
Dialogue: 0,0:08:35.00,0:08:36.13,Default,,0000,0000,0000,,5 times 2 is 10.
Dialogue: 0,0:08:36.13,0:08:38.85,Default,,0000,0000,0000,,So this is equal to 10/11.
Dialogue: 0,0:08:38.85,0:08:39.84,Default,,0000,0000,0000,,One more.
Dialogue: 0,0:08:39.84,0:08:42.06,Default,,0000,0000,0000,,f of z.
Dialogue: 0,0:08:42.06,0:08:43.30,Default,,0000,0000,0000,,Well every time we saw\Nan x, we're going to
Dialogue: 0,0:08:43.30,0:08:44.49,Default,,0000,0000,0000,,replace it with a z.
Dialogue: 0,0:08:44.49,0:08:49.96,Default,,0000,0000,0000,,It's equal to 5 times\N2 minus z over 11.
Dialogue: 0,0:08:49.96,0:08:50.63,Default,,0000,0000,0000,,And that's our answer.
Dialogue: 0,0:08:50.63,0:08:51.91,Default,,0000,0000,0000,,We could distribute the 5.
Dialogue: 0,0:08:51.91,0:08:57.21,Default,,0000,0000,0000,,You could say this is the same\Nthing as 10 minus 5z over 11.
Dialogue: 0,0:08:57.21,0:09:00.26,Default,,0000,0000,0000,,We could even write it in\Nslope-intercept form.
Dialogue: 0,0:09:00.26,0:09:06.00,Default,,0000,0000,0000,,This is the same thing as\Nminus 5/11 z plus 10/11.
Dialogue: 0,0:09:06.00,0:09:06.99,Default,,0000,0000,0000,,These are all equivalent.
Dialogue: 0,0:09:06.99,0:09:10.43,Default,,0000,0000,0000,,But that is what f\Nof z is equal to.
Dialogue: 0,0:09:10.43,0:09:11.59,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:09:11.59,0:09:15.51,Default,,0000,0000,0000,,A function, we said, if you give\Nme any x value, I will
Dialogue: 0,0:09:15.51,0:09:16.47,Default,,0000,0000,0000,,give you an output.
Dialogue: 0,0:09:16.47,0:09:19.12,Default,,0000,0000,0000,,I will give you an f of x.
Dialogue: 0,0:09:19.12,0:09:23.04,Default,,0000,0000,0000,,So if this is our function,\Nyou give me an x, it will
Dialogue: 0,0:09:23.04,0:09:26.55,Default,,0000,0000,0000,,produce an f of x.
Dialogue: 0,0:09:26.55,0:09:29.68,Default,,0000,0000,0000,,It can only produce 1\Nf of x for any x.
Dialogue: 0,0:09:29.68,0:09:32.84,Default,,0000,0000,0000,,You can't have a function that\Ncould produce two possible
Dialogue: 0,0:09:32.84,0:09:34.70,Default,,0000,0000,0000,,values for an x.
Dialogue: 0,0:09:34.70,0:09:37.54,Default,,0000,0000,0000,,So you can't have a function--\Nthis would be an invalid
Dialogue: 0,0:09:37.54,0:09:42.79,Default,,0000,0000,0000,,function definition-- f\Nof x is equal to 3 if
Dialogue: 0,0:09:42.79,0:09:45.23,Default,,0000,0000,0000,,x is equal to 0.
Dialogue: 0,0:09:45.23,0:09:49.24,Default,,0000,0000,0000,,Or it could be equal to\N4 if x is equal to 0.
Dialogue: 0,0:09:49.24,0:09:53.17,Default,,0000,0000,0000,,Because in this situation, we\Ndon't know what f of 0 is.
Dialogue: 0,0:09:53.17,0:09:54.09,Default,,0000,0000,0000,,What it's going to be equal?
Dialogue: 0,0:09:54.09,0:09:56.33,Default,,0000,0000,0000,,It says if x is equal to 0, it\Nshould be 3 or it could be--
Dialogue: 0,0:09:56.33,0:09:57.31,Default,,0000,0000,0000,,we don't know.
Dialogue: 0,0:09:57.31,0:09:57.83,Default,,0000,0000,0000,,We don't know.
Dialogue: 0,0:09:57.83,0:09:58.19,Default,,0000,0000,0000,,We don't know.
Dialogue: 0,0:09:58.19,0:10:01.55,Default,,0000,0000,0000,,This is not a function\Neven though it might
Dialogue: 0,0:10:01.55,0:10:02.80,Default,,0000,0000,0000,,have looked like one.
Dialogue: 0,0:10:07.70,0:10:12.25,Default,,0000,0000,0000,,So you can't have two f of\Nx values for one x value.
Dialogue: 0,0:10:12.25,0:10:16.02,Default,,0000,0000,0000,,So let's see which of these\Ngraphs are functions.
Dialogue: 0,0:10:16.02,0:10:18.39,Default,,0000,0000,0000,,To figure that out, you could\Nsay, look at any x value
Dialogue: 0,0:10:18.39,0:10:21.85,Default,,0000,0000,0000,,here-- pick any x value-- I have\Nexactly one f of x value.
Dialogue: 0,0:10:21.85,0:10:25.09,Default,,0000,0000,0000,,This is y is equal to\Nf of x right here.
Dialogue: 0,0:10:25.09,0:10:28.95,Default,,0000,0000,0000,,I have exactly only one--\Nat that x, that
Dialogue: 0,0:10:28.95,0:10:30.55,Default,,0000,0000,0000,,is my y value here.
Dialogue: 0,0:10:30.55,0:10:32.97,Default,,0000,0000,0000,,So you could have a vertical\Nline test, which says at any
Dialogue: 0,0:10:32.97,0:10:35.72,Default,,0000,0000,0000,,point if you draw a vertical\Nline-- notice a vertical line
Dialogue: 0,0:10:35.72,0:10:37.57,Default,,0000,0000,0000,,is for a certain x value.
Dialogue: 0,0:10:37.57,0:10:41.92,Default,,0000,0000,0000,,That shows that I only have\None y value at that point.
Dialogue: 0,0:10:41.92,0:10:43.63,Default,,0000,0000,0000,,So this is a valid function.
Dialogue: 0,0:10:43.63,0:10:46.24,Default,,0000,0000,0000,,Any time you draw a vertical\Nline, it will only intersect
Dialogue: 0,0:10:46.24,0:10:47.61,Default,,0000,0000,0000,,the graph once.
Dialogue: 0,0:10:47.61,0:10:50.41,Default,,0000,0000,0000,,So this is a valid function.
Dialogue: 0,0:10:50.41,0:10:52.22,Default,,0000,0000,0000,,Now what about this\None right here?
Dialogue: 0,0:10:52.22,0:10:53.96,Default,,0000,0000,0000,,I could draw a vertical\Nline, let's say, at
Dialogue: 0,0:10:53.96,0:10:55.23,Default,,0000,0000,0000,,that point right there.
Dialogue: 0,0:10:55.23,0:10:58.65,Default,,0000,0000,0000,,For that x, this relation\Nseems to have two
Dialogue: 0,0:10:58.65,0:11:00.86,Default,,0000,0000,0000,,possible f of x's.
Dialogue: 0,0:11:00.86,0:11:04.55,Default,,0000,0000,0000,,f of x could be that value or\Nf of x could be that value.
Dialogue: 0,0:11:04.55,0:11:05.27,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:11:05.27,0:11:07.52,Default,,0000,0000,0000,,We're intersecting\Nthe graph twice.
Dialogue: 0,0:11:07.52,0:11:08.84,Default,,0000,0000,0000,,So this is not a function.
Dialogue: 0,0:11:08.84,0:11:11.15,Default,,0000,0000,0000,,We're doing exactly what\NI described here.
Dialogue: 0,0:11:11.15,0:11:15.09,Default,,0000,0000,0000,,For a certain x, we're\Ndescribing two possible y's
Dialogue: 0,0:11:15.09,0:11:16.80,Default,,0000,0000,0000,,that could be equal to f of x.
Dialogue: 0,0:11:16.80,0:11:19.22,Default,,0000,0000,0000,,So this is not a function.
Dialogue: 0,0:11:19.22,0:11:20.83,Default,,0000,0000,0000,,Over here, same thing.
Dialogue: 0,0:11:20.83,0:11:22.31,Default,,0000,0000,0000,,You draw a vertical\Nline right there.
Dialogue: 0,0:11:22.31,0:11:24.54,Default,,0000,0000,0000,,You're intersecting\Nthe graph twice.
Dialogue: 0,0:11:24.54,0:11:26.00,Default,,0000,0000,0000,,This is not a function.
Dialogue: 0,0:11:26.00,0:11:30.59,Default,,0000,0000,0000,,You're defining two possible\Ny values for 1 x value.
Dialogue: 0,0:11:30.59,0:11:31.49,Default,,0000,0000,0000,,Let's go to this function.
Dialogue: 0,0:11:31.49,0:11:33.16,Default,,0000,0000,0000,,It's kind of a weird\Nlooking function.
Dialogue: 0,0:11:33.16,0:11:34.75,Default,,0000,0000,0000,,Looks like a reversed\Ncheck mark.
Dialogue: 0,0:11:34.75,0:11:37.02,Default,,0000,0000,0000,,But any time you draw a vertical\Nline, you're only
Dialogue: 0,0:11:37.02,0:11:38.72,Default,,0000,0000,0000,,intersecting it once.
Dialogue: 0,0:11:38.72,0:11:40.42,Default,,0000,0000,0000,,So this is a valid function.
Dialogue: 0,0:11:40.42,0:11:43.47,Default,,0000,0000,0000,,For every x, you only have\None y associated.
Dialogue: 0,0:11:43.47,0:11:46.45,Default,,0000,0000,0000,,Or only one f of x associated\Nwith it.
Dialogue: 0,0:11:46.45,0:11:48.96,Default,,0000,0000,0000,,Anyway, hopefully you\Nfound that useful.