[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.20,Default,,0000,0000,0000,,- [Instructor] We're told consider
Dialogue: 0,0:00:01.20,0:00:03.63,Default,,0000,0000,0000,,the exponential function f,
Dialogue: 0,0:00:03.63,0:00:05.58,Default,,0000,0000,0000,,which they have to write over here,
Dialogue: 0,0:00:05.58,0:00:06.51,Default,,0000,0000,0000,,what is the domain
Dialogue: 0,0:00:06.51,0:00:08.40,Default,,0000,0000,0000,,and what is the range of f?
Dialogue: 0,0:00:08.40,0:00:11.15,Default,,0000,0000,0000,,So pause this video and see\Nif you can figure that out.
Dialogue: 0,0:00:12.39,0:00:14.04,Default,,0000,0000,0000,,All right, now let's work\Nthrough this together.
Dialogue: 0,0:00:14.04,0:00:16.14,Default,,0000,0000,0000,,So let's, first of all,\Njust remind ourselves
Dialogue: 0,0:00:16.14,0:00:18.57,Default,,0000,0000,0000,,what domain and range mean.
Dialogue: 0,0:00:18.57,0:00:22.50,Default,,0000,0000,0000,,Domain is all of the x values
Dialogue: 0,0:00:22.50,0:00:24.99,Default,,0000,0000,0000,,that we could input into our function
Dialogue: 0,0:00:24.99,0:00:27.66,Default,,0000,0000,0000,,where our function is defined.
Dialogue: 0,0:00:27.66,0:00:29.10,Default,,0000,0000,0000,,So if we look over here,
Dialogue: 0,0:00:29.10,0:00:32.37,Default,,0000,0000,0000,,it looks like we can\Ntake any real number x
Dialogue: 0,0:00:32.37,0:00:34.71,Default,,0000,0000,0000,,that is any positive value.
Dialogue: 0,0:00:34.71,0:00:35.67,Default,,0000,0000,0000,,It looks like it's defined.
Dialogue: 0,0:00:35.67,0:00:37.86,Default,,0000,0000,0000,,This graph keeps going\Non and on to the right,
Dialogue: 0,0:00:37.86,0:00:40.08,Default,,0000,0000,0000,,and this graph keeps going\Non and on to the left.
Dialogue: 0,0:00:40.08,0:00:41.70,Default,,0000,0000,0000,,We could also take on negative values.
Dialogue: 0,0:00:41.70,0:00:43.36,Default,,0000,0000,0000,,We could even say x equals 0.
Dialogue: 0,0:00:43.36,0:00:45.50,Default,,0000,0000,0000,,I don't see any gaps here
Dialogue: 0,0:00:45.50,0:00:48.21,Default,,0000,0000,0000,,where our function is not defined.
Dialogue: 0,0:00:48.21,0:00:50.01,Default,,0000,0000,0000,,So our domain looks like
Dialogue: 0,0:00:50.01,0:00:53.97,Default,,0000,0000,0000,,all real numbers.
Dialogue: 0,0:00:53.97,0:00:55.32,Default,,0000,0000,0000,,Or another way to think about it
Dialogue: 0,0:00:55.32,0:00:58.59,Default,,0000,0000,0000,,is x can take on any real number.
Dialogue: 0,0:00:58.59,0:01:00.09,Default,,0000,0000,0000,,And if you put it into our function,
Dialogue: 0,0:01:00.09,0:01:02.37,Default,,0000,0000,0000,,f of x is going to be defined.
Dialogue: 0,0:01:02.37,0:01:05.67,Default,,0000,0000,0000,,So now let's think about the range.
Dialogue: 0,0:01:05.67,0:01:07.98,Default,,0000,0000,0000,,The range, as a reminder,
Dialogue: 0,0:01:07.98,0:01:11.07,Default,,0000,0000,0000,,is the set of all of the values
Dialogue: 0,0:01:11.07,0:01:14.10,Default,,0000,0000,0000,,that our function can take on.
Dialogue: 0,0:01:14.10,0:01:16.74,Default,,0000,0000,0000,,So when we look at this over here,
Dialogue: 0,0:01:16.74,0:01:18.54,Default,,0000,0000,0000,,it looks like if our x values
Dialogue: 0,0:01:18.54,0:01:20.28,Default,,0000,0000,0000,,get more and more negative,
Dialogue: 0,0:01:20.28,0:01:21.87,Default,,0000,0000,0000,,or the value of our function
Dialogue: 0,0:01:21.87,0:01:23.67,Default,,0000,0000,0000,,just goes up towards infinity,
Dialogue: 0,0:01:23.67,0:01:27.12,Default,,0000,0000,0000,,so it can take on these\Narbitrarily large values.
Dialogue: 0,0:01:27.12,0:01:30.45,Default,,0000,0000,0000,,But then as we move in\Nthe positive x direction,
Dialogue: 0,0:01:30.45,0:01:34.59,Default,,0000,0000,0000,,our function value gets\Nlower and lower and lower.
Dialogue: 0,0:01:34.59,0:01:37.11,Default,,0000,0000,0000,,And it looks like it approaches 0,
Dialogue: 0,0:01:37.11,0:01:38.67,Default,,0000,0000,0000,,but never quite gets to 0.
Dialogue: 0,0:01:38.67,0:01:41.04,Default,,0000,0000,0000,,And actually that's what this dotted line
Dialogue: 0,0:01:41.04,0:01:42.24,Default,,0000,0000,0000,,over here represents.
Dialogue: 0,0:01:42.24,0:01:43.38,Default,,0000,0000,0000,,That's an asymptote.
Dialogue: 0,0:01:43.38,0:01:46.77,Default,,0000,0000,0000,,That means that as x gets\Nlarger and larger and larger,
Dialogue: 0,0:01:46.77,0:01:48.66,Default,,0000,0000,0000,,the value of our function is going to get
Dialogue: 0,0:01:48.66,0:01:50.79,Default,,0000,0000,0000,,closer and closer to this dotted line,
Dialogue: 0,0:01:50.79,0:01:53.91,Default,,0000,0000,0000,,which is at y equals 0
Dialogue: 0,0:01:53.91,0:01:55.98,Default,,0000,0000,0000,,but it never quite gets there.
Dialogue: 0,0:01:55.98,0:01:57.93,Default,,0000,0000,0000,,So it looks like this function
Dialogue: 0,0:01:57.93,0:02:00.90,Default,,0000,0000,0000,,can take on the any real value
Dialogue: 0,0:02:00.90,0:02:02.88,Default,,0000,0000,0000,,that is greater than 0,
Dialogue: 0,0:02:02.88,0:02:05.31,Default,,0000,0000,0000,,but not at 0 or below 0.
Dialogue: 0,0:02:05.31,0:02:09.13,Default,,0000,0000,0000,,So all real numbers
Dialogue: 0,0:02:10.98,0:02:15.63,Default,,0000,0000,0000,,greater than, greater than 0.
Dialogue: 0,0:02:15.63,0:02:17.10,Default,,0000,0000,0000,,Or another way to think about it
Dialogue: 0,0:02:17.10,0:02:18.93,Default,,0000,0000,0000,,is we could set the range of saying
Dialogue: 0,0:02:18.93,0:02:22.23,Default,,0000,0000,0000,,f of x is greater than 0,
Dialogue: 0,0:02:22.23,0:02:23.28,Default,,0000,0000,0000,,not greater than an equal to,
Dialogue: 0,0:02:23.28,0:02:24.33,Default,,0000,0000,0000,,it'll get closer and closer
Dialogue: 0,0:02:24.33,0:02:26.25,Default,,0000,0000,0000,,but not quite equal that.
Dialogue: 0,0:02:26.25,0:02:27.45,Default,,0000,0000,0000,,Let's do another example
Dialogue: 0,0:02:27.45,0:02:31.05,Default,,0000,0000,0000,,where they haven't drawn the graph for us.
Dialogue: 0,0:02:31.05,0:02:33.69,Default,,0000,0000,0000,,So let's look at this one over here.
Dialogue: 0,0:02:33.69,0:02:36.93,Default,,0000,0000,0000,,So consider the exponential function h,
Dialogue: 0,0:02:36.93,0:02:38.31,Default,,0000,0000,0000,,and actually let me get rid of all of this
Dialogue: 0,0:02:38.31,0:02:42.66,Default,,0000,0000,0000,,so that we can focus\Non this actual problem.
Dialogue: 0,0:02:42.66,0:02:44.79,Default,,0000,0000,0000,,So consider the exponential function h
Dialogue: 0,0:02:44.79,0:02:46.35,Default,,0000,0000,0000,,where h of x is equal to that.
Dialogue: 0,0:02:46.35,0:02:49.05,Default,,0000,0000,0000,,What is the domain and\Nwhat is the range of h?
Dialogue: 0,0:02:49.05,0:02:51.51,Default,,0000,0000,0000,,So let's start with the domain.
Dialogue: 0,0:02:52.62,0:02:54.48,Default,,0000,0000,0000,,What are all of the x values
Dialogue: 0,0:02:54.48,0:02:57.45,Default,,0000,0000,0000,,where h of x is defined?
Dialogue: 0,0:02:57.45,0:02:59.22,Default,,0000,0000,0000,,Well, I could put any x value here.
Dialogue: 0,0:02:59.22,0:03:00.66,Default,,0000,0000,0000,,I could put any negative value.
Dialogue: 0,0:03:00.66,0:03:03.09,Default,,0000,0000,0000,,I could say what happens when x equals 0.
Dialogue: 0,0:03:03.09,0:03:05.07,Default,,0000,0000,0000,,I can say any positive value.
Dialogue: 0,0:03:05.07,0:03:06.30,Default,,0000,0000,0000,,So once again, our domain
Dialogue: 0,0:03:06.30,0:03:11.28,Default,,0000,0000,0000,,is all real numbers for x.
Dialogue: 0,0:03:11.28,0:03:12.78,Default,,0000,0000,0000,,Now what about our range?
Dialogue: 0,0:03:12.78,0:03:14.19,Default,,0000,0000,0000,,This one is interesting.
Dialogue: 0,0:03:15.03,0:03:18.78,Default,,0000,0000,0000,,What happens when x gets\Nreally, really, really large?
Dialogue: 0,0:03:18.78,0:03:19.93,Default,,0000,0000,0000,,Let's pick a large x.
Dialogue: 0,0:03:19.93,0:03:23.76,Default,,0000,0000,0000,,Let's say we're thinking about h of 30,
Dialogue: 0,0:03:23.76,0:03:25.02,Default,,0000,0000,0000,,which isn't even that large,
Dialogue: 0,0:03:25.02,0:03:26.04,Default,,0000,0000,0000,,but let's think about what happens.
Dialogue: 0,0:03:26.04,0:03:31.04,Default,,0000,0000,0000,,That's -7 times 2/3 to the 30th power.
Dialogue: 0,0:03:31.23,0:03:33.27,Default,,0000,0000,0000,,What does 2/3 to the 30th power look like?
Dialogue: 0,0:03:33.27,0:03:35.73,Default,,0000,0000,0000,,That's the same thing as equal to -7
Dialogue: 0,0:03:35.73,0:03:37.92,Default,,0000,0000,0000,,times 2 to the 30th
Dialogue: 0,0:03:37.92,0:03:39.69,Default,,0000,0000,0000,,over 3 to the 30th.
Dialogue: 0,0:03:39.69,0:03:40.68,Default,,0000,0000,0000,,You might not realize it,
Dialogue: 0,0:03:40.68,0:03:43.80,Default,,0000,0000,0000,,but 3 to the 30th is much\Nlarger than 2 to the 30th.
Dialogue: 0,0:03:43.80,0:03:45.42,Default,,0000,0000,0000,,This number right over here
Dialogue: 0,0:03:45.42,0:03:48.42,Default,,0000,0000,0000,,is awfully close to 0.
Dialogue: 0,0:03:48.42,0:03:49.98,Default,,0000,0000,0000,,In fact, if you want to verify that,
Dialogue: 0,0:03:49.98,0:03:51.45,Default,,0000,0000,0000,,lemme take a calculator out
Dialogue: 0,0:03:51.45,0:03:52.65,Default,,0000,0000,0000,,and I could show you that.
Dialogue: 0,0:03:52.65,0:03:55.41,Default,,0000,0000,0000,,If I took 2 divided by 3
Dialogue: 0,0:03:55.41,0:03:57.15,Default,,0000,0000,0000,,which we know is .6 repeating,
Dialogue: 0,0:03:57.15,0:03:59.25,Default,,0000,0000,0000,,and if I were to take that
Dialogue: 0,0:03:59.25,0:04:00.69,Default,,0000,0000,0000,,to the 30th power,
Dialogue: 0,0:04:01.56,0:04:05.52,Default,,0000,0000,0000,,it equals a very, very,\Nvery small positive number.
Dialogue: 0,0:04:05.52,0:04:08.46,Default,,0000,0000,0000,,But then, we're going to\Nmultiply that times -7.
Dialogue: 0,0:04:08.46,0:04:09.45,Default,,0000,0000,0000,,And if we want, let's do that,
Dialogue: 0,0:04:09.45,0:04:10.57,Default,,0000,0000,0000,,times -7.
Dialogue: 0,0:04:13.95,0:04:18.78,Default,,0000,0000,0000,,It equals a very, very\Nsmall negative number.
Dialogue: 0,0:04:18.78,0:04:20.67,Default,,0000,0000,0000,,Now if you go the other way,
Dialogue: 0,0:04:20.67,0:04:22.44,Default,,0000,0000,0000,,if you think about negative exponents,
Dialogue: 0,0:04:22.44,0:04:26.01,Default,,0000,0000,0000,,so let's say we have h of -30,
Dialogue: 0,0:04:26.01,0:04:30.46,Default,,0000,0000,0000,,that's going to be -7\Ntimes 2/3 to the -30,
Dialogue: 0,0:04:30.46,0:04:33.33,Default,,0000,0000,0000,,which is the same thing as -7;
Dialogue: 0,0:04:33.33,0:04:35.25,Default,,0000,0000,0000,,this negative, instead of\Nit writing it that way,
Dialogue: 0,0:04:35.25,0:04:36.69,Default,,0000,0000,0000,,we could take the reciprocal here,
Dialogue: 0,0:04:36.69,0:04:38.61,Default,,0000,0000,0000,,this is the same thing as 3/2
Dialogue: 0,0:04:38.61,0:04:40.38,Default,,0000,0000,0000,,to the +30 power.
Dialogue: 0,0:04:40.38,0:04:43.50,Default,,0000,0000,0000,,Now this is a very large positive number,
Dialogue: 0,0:04:43.50,0:04:45.57,Default,,0000,0000,0000,,which we will then multiply by -7
Dialogue: 0,0:04:45.57,0:04:47.85,Default,,0000,0000,0000,,to get a very large negative number.
Dialogue: 0,0:04:47.85,0:04:48.69,Default,,0000,0000,0000,,Just to show you
Dialogue: 0,0:04:48.69,0:04:52.77,Default,,0000,0000,0000,,that that is a very large positive number,
Dialogue: 0,0:04:52.77,0:04:54.57,Default,,0000,0000,0000,,so if I take 3/2,
Dialogue: 0,0:04:54.57,0:04:57.24,Default,,0000,0000,0000,,which is 1.5 of course, 3/2,
Dialogue: 0,0:04:57.24,0:04:59.28,Default,,0000,0000,0000,,and I am going to raise that
Dialogue: 0,0:04:59.28,0:05:00.69,Default,,0000,0000,0000,,to the 30th power,
Dialogue: 0,0:05:01.98,0:05:05.37,Default,,0000,0000,0000,,well, it's roughly 192,000.
Dialogue: 0,0:05:05.37,0:05:06.76,Default,,0000,0000,0000,,But now, if I multiply by -7,
Dialogue: 0,0:05:06.76,0:05:11.76,Default,,0000,0000,0000,,it's gonna become a large\Nnegative number, times -7,
Dialogue: 0,0:05:11.88,0:05:15.69,Default,,0000,0000,0000,,it's equal to a little\Nbit over negative million.
Dialogue: 0,0:05:15.69,0:05:18.36,Default,,0000,0000,0000,,So one way to visualize this graph,
Dialogue: 0,0:05:18.36,0:05:20.22,Default,,0000,0000,0000,,and I'll do it very quickly,
Dialogue: 0,0:05:20.22,0:05:21.81,Default,,0000,0000,0000,,is what's happening here.
Dialogue: 0,0:05:23.07,0:05:23.90,Default,,0000,0000,0000,,And if we want,
Dialogue: 0,0:05:23.90,0:05:25.34,Default,,0000,0000,0000,,we can think about if this is the x axis,
Dialogue: 0,0:05:25.34,0:05:26.88,Default,,0000,0000,0000,,this is the y axis,
Dialogue: 0,0:05:26.88,0:05:29.01,Default,,0000,0000,0000,,we can even think about when x equals 0,
Dialogue: 0,0:05:29.01,0:05:30.36,Default,,0000,0000,0000,,this is all 1.
Dialogue: 0,0:05:30.36,0:05:33.09,Default,,0000,0000,0000,,And so h of 0 is equal to -7,
Dialogue: 0,0:05:33.09,0:05:36.18,Default,,0000,0000,0000,,so if we say -7 right over here.
Dialogue: 0,0:05:36.18,0:05:38.55,Default,,0000,0000,0000,,When x is very negative,
Dialogue: 0,0:05:38.55,0:05:40.92,Default,,0000,0000,0000,,h takes on very large negative values,
Dialogue: 0,0:05:40.92,0:05:42.45,Default,,0000,0000,0000,,we just saw that.
Dialogue: 0,0:05:42.45,0:05:45.18,Default,,0000,0000,0000,,And then as x becomes\Nmore and more positive,
Dialogue: 0,0:05:45.18,0:05:47.64,Default,,0000,0000,0000,,it approaches 0.
Dialogue: 0,0:05:47.64,0:05:49.50,Default,,0000,0000,0000,,The function approaches 0
Dialogue: 0,0:05:49.50,0:05:52.08,Default,,0000,0000,0000,,but never quite exactly gets there.
Dialogue: 0,0:05:52.08,0:05:52.91,Default,,0000,0000,0000,,And so once again,
Dialogue: 0,0:05:52.91,0:05:54.18,Default,,0000,0000,0000,,we could draw that dotted,
Dialogue: 0,0:05:54.18,0:05:56.07,Default,,0000,0000,0000,,lemme do that in a different\Ncolor so you can see it,
Dialogue: 0,0:05:56.07,0:05:59.97,Default,,0000,0000,0000,,we can draw that dotted\Nasymptote line right over there.
Dialogue: 0,0:05:59.97,0:06:01.50,Default,,0000,0000,0000,,So what's the range?
Dialogue: 0,0:06:01.50,0:06:04.50,Default,,0000,0000,0000,,So we could say all real numbers
Dialogue: 0,0:06:04.50,0:06:06.33,Default,,0000,0000,0000,,less than 0.
Dialogue: 0,0:06:06.33,0:06:08.34,Default,,0000,0000,0000,,So let me write that, it is:
Dialogue: 0,0:06:10.89,0:06:13.81,Default,,0000,0000,0000,,all real numbers
Dialogue: 0,0:06:15.15,0:06:18.81,Default,,0000,0000,0000,,less than 0.
Dialogue: 0,0:06:18.81,0:06:21.18,Default,,0000,0000,0000,,Or we could say that f of x
Dialogue: 0,0:06:21.18,0:06:23.43,Default,,0000,0000,0000,,can take on any value less than 0:
Dialogue: 0,0:06:23.43,0:06:25.38,Default,,0000,0000,0000,,f of x is going to be less than 0.
Dialogue: 0,0:06:25.38,0:06:27.75,Default,,0000,0000,0000,,It approaches 0 as x\Ngets larger and larger
Dialogue: 0,0:06:27.75,0:06:29.55,Default,,0000,0000,0000,,but never quite gets there.