Video: Finding the Range of an Exponential Function

Find the range of the function 𝑓(π‘₯) = 10^π‘₯.

03:18

Video Transcript

Find the range of the function 𝑓 of π‘₯ equals 10 to the π‘₯ power.

The first thing we need to consider is what the range of a function is. It’s the set of all possible resulting values of the dependent variable. When we look at 𝑓 of π‘₯ equals 10 to the π‘₯ power, the range will be whatever the 𝑓 of π‘₯ can be, its possible resulting values. And to answer that, we need to think about the nature of exponents.

We know that if we have π‘Ž to the negative π‘₯ power, that means that we have a negative exponent. It will be equal to one over π‘Ž to the π‘₯ power. A negative exponent is a fractional value. Let’s consider what would happen if we plugged in negative two for our exponent, for our π‘₯-value. The 𝑓 of π‘₯, the output, would be one over 10 squared. We can say that when π‘₯ is negative, the 𝑓 of π‘₯ is positive. We could also consider a positive value for π‘₯, like 10 squared. Then the 𝑓 of π‘₯ would be equal to 100. We can say that when π‘₯ is positive, 𝑓 of π‘₯ is still positive.

For this equation, there is no place where the 𝑓 of π‘₯ would be negative. At zero, when π‘₯ equals zero, 10 to the zero power equals one, still positive. If we sketch a graph of this kind of exponential function, we have our π‘₯- and 𝑦-axis. And then we’d have a line that looks something like this. As we move from left to right on the π‘₯-axis, our 𝑦-values get larger and larger. As we move to the left on the π‘₯-axis, these are our negative π‘₯-values, the function is getting smaller and smaller. But remember, it’s never going to be negative. And that means it’s never going to cross below the π‘₯-axis.

And we want to represent this in set notation. We can say that our 𝑓 of π‘₯ is always larger than zero. But there is no upper limit to our 𝑓 of π‘₯. It continues to increase. So, we say that it increases to infinity. This is the set notation of 𝑓 of π‘₯ being between zero and infinity.

Notice that the brackets face outwards. This is because our 𝑓 of π‘₯ falls between these two values, but it is not equal to either one. If, for example, you flipped this bracket around before writing the zero, that would mean that 𝑓 of π‘₯ could be equal to zero. But there is no place where 𝑓 of π‘₯ is equal to zero. We have to exclude that value.

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