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- [Instructor] We're told consider

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the exponential function f,

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which they have to write over here,

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what is the domain

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and what is the range of f?

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So pause this video and see[br]if you can figure that out.

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All right, now let's work[br]through this together.

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So let's, first of all,[br]just remind ourselves

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what domain and range mean.

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Domain is all of the x values

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that we could input into our function

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where our function is defined.

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So if we look over here,

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it looks like we can[br]take any real number x

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that is any positive value.

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It looks like it's defined.

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This graph keeps going[br]on and on to the right,

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and this graph keeps going[br]on and on to the left.

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We could also take on negative values.

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We could even say x equals 0.

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I don't see any gaps here

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where our function is not defined.

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So our domain looks like

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all real numbers.

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Or another way to think about it

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is x can take on any real number.

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And if you put it into our function,

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f of x is going to be defined.

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So now let's think about the range.

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The range, as a reminder,

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is the set of all of the values

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that our function can take on.

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So when we look at this over here,

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it looks like if our x values

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get more and more negative,

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or the value of our function

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just goes up towards infinity,

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so it can take on these[br]arbitrarily large values.

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But then as we move in[br]the positive x direction,

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our function value gets[br]lower and lower and lower.

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And it looks like it approaches 0,

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but never quite gets to 0.

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And actually that's what this dotted line

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over here represents.

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That's an asymptote.

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That means that as x gets[br]larger and larger and larger,

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the value of our function is going to get

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closer and closer to this dotted line,

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which is at y equals 0

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but it never quite gets there.

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So it looks like this function

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can take on the any real value

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that is greater than 0,

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but not at 0 or below 0.

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So all real numbers

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greater than, greater than 0.

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Or another way to think about it

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is we could set the range of saying

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f of x is greater than 0,

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not greater than an equal to,

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it'll get closer and closer

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but not quite equal that.

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Let's do another example

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where they haven't drawn the graph for us.

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So let's look at this one over here.

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So consider the exponential function h,

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and actually let me get rid of all of this

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so that we can focus[br]on this actual problem.

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So consider the exponential function h

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where h of x is equal to that.

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What is the domain and[br]what is the range of h?

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So let's start with the domain.

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What are all of the x values

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where h of x is defined?

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Well, I could put any x value here.

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I could put any negative value.

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I could say what happens when x equals 0.

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I can say any positive value.

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So once again, our domain

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is all real numbers for x.

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Now what about our range?

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This one is interesting.

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What happens when x gets[br]really, really, really large?

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Let's pick a large x.

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Let's say we're thinking about h of 30,

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which isn't even that large,

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but let's think about what happens.

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That's -7 times 2/3 to the 30th power.

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What does 2/3 to the 30th power look like?

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That's the same thing as equal to -7

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times 2 to the 30th

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over 3 to the 30th.

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You might not realize it,

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but 3 to the 30th is much[br]larger than 2 to the 30th.

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This number right over here

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is awfully close to 0.

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In fact, if you want to verify that,

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lemme take a calculator out

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and I could show you that.

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If I took 2 divided by 3

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which we know is .6 repeating,

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and if I were to take that

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to the 30th power,

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it equals a very, very,[br]very small positive number.

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But then, we're going to[br]multiply that times -7.

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And if we want, let's do that,

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times -7.

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It equals a very, very[br]small negative number.

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Now if you go the other way,

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if you think about negative exponents,

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so let's say we have h of -30,

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that's going to be -7[br]times 2/3 to the -30,

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which is the same thing as -7;

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this negative, instead of[br]it writing it that way,

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we could take the reciprocal here,

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this is the same thing as 3/2

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to the +30 power.

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Now this is a very large positive number,

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which we will then multiply by -7

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to get a very large negative number.

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Just to show you

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that that is a very large positive number,

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so if I take 3/2,

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which is 1.5 of course, 3/2,

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and I am going to raise that

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to the 30th power,

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well, it's roughly 192,000.

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But now, if I multiply by -7,

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it's gonna become a large[br]negative number, times -7,

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it's equal to a little[br]bit over negative million.

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So one way to visualize this graph,

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and I'll do it very quickly,

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is what's happening here.

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And if we want,

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we can think about if this is the x axis,

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this is the y axis,

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we can even think about when x equals 0,

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this is all 1.

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And so h of 0 is equal to -7,

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so if we say -7 right over here.

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When x is very negative,

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h takes on very large negative values,

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we just saw that.

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And then as x becomes[br]more and more positive,

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it approaches 0.

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The function approaches 0

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but never quite exactly gets there.

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And so once again,

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we could draw that dotted,

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lemme do that in a different[br]color so you can see it,

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we can draw that dotted[br]asymptote line right over there.

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So what's the range?

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So we could say all real numbers

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less than 0.

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So let me write that, it is:

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all real numbers

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less than 0.

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Or we could say that f of x

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can take on any value less than 0:

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f of x is going to be less than 0.

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It approaches 0 as x[br]gets larger and larger

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but never quite gets there.