Determining and representing the domain and range of exponential functions | Khan Academy

- [Instructor] We're told consider

the exponential function f,

which they have to write over here,

what is the domain

and what is the range of f?

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

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

So let's, first of all,
just remind ourselves

what domain and range mean.

Domain is all of the x values

that we could input into our function

where our function is defined.

So if we look over here,

it looks like we can
take any real number x

that is any positive value.

It looks like it's defined.

This graph keeps going
on and on to the right,

and this graph keeps going
on and on to the left.

We could also take on negative values.

We could even say x equals 0.

I don't see any gaps here

where our function is not defined.

So our domain looks like

all real numbers.

Or another way to think about it

is x can take on any real number.

And if you put it into our function,

f of x is going to be defined.

So now let's think about the range.

The range, as a reminder,

is the set of all of the values

that our function can take on.

So when we look at this over here,

it looks like if our x values

get more and more negative,

or the value of our function

just goes up towards infinity,

so it can take on these
arbitrarily large values.

But then as we move in
the positive x direction,

our function value gets
lower and lower and lower.

And it looks like it approaches 0,

but never quite gets to 0.

And actually that's what this dotted line

over here represents.

That's an asymptote.

That means that as x gets
larger and larger and larger,

the value of our function is going to get

closer and closer to this dotted line,

which is at y equals 0

but it never quite gets there.

So it looks like this function

can take on the any real value

that is greater than 0,

but not at 0 or below 0.

So all real numbers

greater than, greater than 0.

Or another way to think about it

is we could set the range of saying

f of x is greater than 0,

not greater than an equal to,

it'll get closer and closer

but not quite equal that.

Let's do another example

where they haven't drawn the graph for us.

So let's look at this one over here.

So consider the exponential function h,

and actually let me get rid of all of this

so that we can focus
on this actual problem.

So consider the exponential function h

where h of x is equal to that.

What is the domain and
what is the range of h?

So let's start with the domain.

What are all of the x values

where h of x is defined?

Well, I could put any x value here.

I could put any negative value.

I could say what happens when x equals 0.

I can say any positive value.

So once again, our domain

is all real numbers for x.

Now what about our range?

This one is interesting.

What happens when x gets
really, really, really large?

Let's pick a large x.

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

which isn't even that large,

but let's think about what happens.

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

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

That's the same thing as equal to -7

times 2 to the 30th

over 3 to the 30th.

You might not realize it,

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

This number right over here

is awfully close to 0.

In fact, if you want to verify that,

lemme take a calculator out

and I could show you that.

If I took 2 divided by 3

which we know is .6 repeating,

and if I were to take that

to the 30th power,

it equals a very, very,
very small positive number.

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

And if we want, let's do that,

times -7.

It equals a very, very
small negative number.

Now if you go the other way,

if you think about negative exponents,

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

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

which is the same thing as -7;

this negative, instead of
it writing it that way,

we could take the reciprocal here,

this is the same thing as 3/2

to the +30 power.

Now this is a very large positive number,

which we will then multiply by -7

to get a very large negative number.

Just to show you

that that is a very large positive number,

so if I take 3/2,

which is 1.5 of course, 3/2,

and I am going to raise that

to the 30th power,

well, it's roughly 192,000.

But now, if I multiply by -7,

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

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

So one way to visualize this graph,

and I'll do it very quickly,

is what's happening here.

And if we want,

we can think about if this is the x axis,

this is the y axis,

we can even think about when x equals 0,

this is all 1.

And so h of 0 is equal to -7,

so if we say -7 right over here.

When x is very negative,

h takes on very large negative values,

we just saw that.

And then as x becomes
more and more positive,

it approaches 0.

The function approaches 0

but never quite exactly gets there.

And so once again,

we could draw that dotted,

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

we can draw that dotted
asymptote line right over there.

So what's the range?

So we could say all real numbers

less than 0.

So let me write that, it is:

all real numbers

less than 0.

Or we could say that f of x

can take on any value less than 0:

f of x is going to be less than 0.

It approaches 0 as x
gets larger and larger

but never quite gets there.