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