Find functions satisfying the given conditions in each of the following cases.

- for and .
- for all , and .
- for all and .
- and .

- We make the substitution . Since this gives us . Therefore,
Since we are given we can solve for ,

Therefore,

- We make the substitution . Since we then have . Therefore,
Since we are given that we can solve for ,

Therefore,

This formula is valid for since and for all .

- We make the substitution . Since we then have . So,
Since we are given that we can solve for ,

Therefore,

This is valid for since and for all .

- We make the substitution . Then, and so we have
So, we consider the two cases separately. If then we have and

If then we have and

Therefore, we have the function

Now, to solve for we use the condition that . (Here we’re going to assume we want to make the function continuous at , i.e., that the two pieces of this piecewise definition take the same value at 0 so that the limits from the left and right would be equal.) Therefore, we have

and

Thus, the function is given by