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

We're told to consider the exponential function f, which they've after righted 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 z. 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. If you put it into our function, f of x is going to be defined.

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, 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 zero but never quite gets to zero.

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 equal 0, but it never quite gets there. So it looks like this function can take on any real value that is greater than zero, but not at zero or below zero.

So all real numbers greater than zero, or another way to think about it is we could set the range of saying f of x is greater than zero, not greater than or equal to. It'll get closer and closer but not quite equal.

Let's do another example where they haven't drawn the graph for us. 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 z. 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 * (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 * (2^30) / (3^30). You might not realize it, but (3^30) is much larger than (2^30). This number right over here is awfully close to zero.

In fact, if you want to verify that, let me take a calculator out, and I could show you that if I took (2 / 3), which we know is 0.666 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 * (2/3) to the -30, which is the same thing as -7. This negative, instead of writing it that way, we could take the reciprocal here. This is the same thing as -7 * (3/2) to the positive 30th 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, that is a, well, it's roughly 192,000. But now if I multiply by -7, it's going to become a large negative number.

Just to show you that is a very large positive number. If we take three halves, which is 1.5, and I am going to raise that to the 30th power, that is, well, it's roughly 192,000.

But now if I multiply by -7, it's going to become a large negative number times -7. It's equal to a little bit over a million.

So one way to visualize this graph, and I'll do it very quickly, is what's happening here. If we want, we can think about this as the x-axis and the y-axis. We can even think about when x equals 0; this is all one.

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 zero. The function approaches zero but never quite exactly gets there.

And so once again, we could draw that dotted, let me 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 zero. So let me write that: it is all real numbers less than zero, or we could say that f of x can take on any value less than zero. f of x is going to be less than zero. It approaches zero as X gets larger and larger but never quite gets there.