### Video Transcript

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

The first thing we need to consider is what the range of a function is. Itβs the set of all possible resulting values of the dependent variable. When we look at π of π₯ equals 10 to the π₯ power, the range will be whatever the π 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 to the negative π₯ power, that means that we have a negative exponent. It will be equal to one over π to the π₯ power. A negative exponent is a fractional value. Letβs consider what would happen if we plugged in negative two for our exponent, for our π₯-value. The π of π₯, the output, would be one over 10 squared. We can say that when π₯ is negative, the π of π₯ is positive. We could also consider a positive value for π₯, like 10 squared. Then the π of π₯ would be equal to 100. We can say that when π₯ is positive, π of π₯ is still positive.

For this equation, there is no place where the π 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 π¦-axis. And then weβd have a line that looks something like this. As we move from left to right on the π₯-axis, our π¦-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, itβs never going to be negative. And that means itβs never going to cross below the π₯-axis.

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

Notice that the brackets face outwards. This is because our π 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 π of π₯ could be equal to zero. But there is no place where π of π₯ is equal to zero. We have to exclude that value.