The first week of school I gave this problem and never came back to it:

Find the range of .

The answer is . Here’s why:

First consider the range of the quadratic in the exponent, . It’s a parabola that opens up with its vertex at . So the range of is .

Now, consider the function . If we let the domain of be the values coming from the range of , we have the mapping . That is, we’re considering the composition . Since is monotonically increasing, for any , we know . So the range of in is . So the range of is .

Do you feel “at home on the range”?

Here are a few more for you to try. In each case, find the range of the function. These aren’t meant to be any harder than the original problem, just different. Though watch out for the third one.