Find the range of the function 𝑓 of 𝑥 equals 10 to the 𝑥 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 𝑥 equals 10 to the 𝑥 power, the range will be whatever the 𝑓 of 𝑥 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 𝑎 to the negative 𝑥 power, that means that we have a negative exponent. It will be equal to one over 𝑎 to the 𝑥 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 𝑥-value. The 𝑓 of 𝑥, the output, would be one over 10 squared. We can say that when 𝑥 is negative, the 𝑓 of 𝑥 is positive. We could also consider a positive value for 𝑥, like 10 squared. Then the 𝑓 of 𝑥 would be equal to 100. We can say that when 𝑥 is positive, 𝑓 of 𝑥 is still positive.
For this equation, there is no place where the 𝑓 of 𝑥 would be negative. At zero, when 𝑥 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 𝑥- and 𝑦-axis. And then we’d have a line that looks something like this. As we move from left to right on the 𝑥-axis, our 𝑦-values get larger and larger. As we move to the left on the 𝑥-axis, these are our negative 𝑥-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 𝑥-axis.
And we want to represent this in set notation. We can say that our 𝑓 of 𝑥 is always larger than zero. But there is no upper limit to our 𝑓 of 𝑥. It continues to increase. So, we say that it increases to infinity. This is the set notation of 𝑓 of 𝑥 being between zero and infinity.
Notice that the brackets face outwards. This is because our 𝑓 of 𝑥 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 𝑥 could be equal to zero. But there is no place where 𝑓 of 𝑥 is equal to zero. We have to exclude that value.