Problem: Let and
We will explore the graphs of our new functions, and explore the changes in the domains and ranges.
First we consider the graphs of and .
We see immediately, that and .
We notice that appears to remain to the right of the y-axis, and appears to remain above the x-axis.
Just a little thought reveals that in fact this must be the case since for all , This is due to the fact that , and so any power of must also be greater than zero.
Now, for , we must first note how and are related. When we talk about , we are really just asking what power of is equal to . That is, when
, what is ?
For example, since implies that This explains why is always to the right of the y-axis, since for all . Hence, is not defined for .
We say that the domain of , denoted and the range of , denoted
Similarly, and .
Now, , like , is always to the right of the y-axis.
Well, obviously if , since is not defined for .
Hence, and .
We also not that lies between the graphs of and , but this makes sense because we are simply adding the two respective functions to form .
Now consider .
Again we see that and .
But this time the growth of eventually overtakes . Just a little thought we understand why this must be the case, since for all , and if .
Well, as expected and .
What can we guess about and , where .
This time and .
Notice: appears to be very similar to the graph . Well, we need only note that , and so , the identity function.
In this case, we say that and , and so .