Given functions and
with the following values
Let and
and construct a table as above for the functions
and
.
First, by the chain rule we have
Using these formulas we compute the table,
Given functions and
with the following values
Let and
and construct a table as above for the functions
and
.
First, by the chain rule we have
Using these formulas we compute the table,
Define functions and
as follows:
Find a formula for . Find the values at which
is continuous.
If , we have
and we compute the composition,
If , then
and we compute the composition,
Hence,
This is the same function as in this previous exercise, and it is continuous everywhere.
Define functions and
as follows:
Find a formula for and find the values at which
is continuous.
If then
and so
. Therefore, by the definition of
,
If , then
and so again
Next, if then
and so
Finally, if , then
and we have
Putting this all together we have
From this expression we have is continuous everywhere except at
and
.
Define functions and
as follows:
Find a formula for . Find the values at which
is continuous.
If , we have
and we compute the composition,
If , then
and we compute the composition,
Putting these together,
Certainly is continuous for all
since the constant function
and the polynomial
are continuous. Further,
is continuous at
since
Find the composition function of the functions:
We compute ,
This is valid for all . (Of course, this formula would generally be valid for
as well, but since the domain of
and
is given as
, the composition is defined only for
.)
Find the composition function of the functions:
We compute ,
This is valid for all . (Of course, this formula would generally be valid for
as well, but since the domain of
and
is given as
, the composition is defined only for
.)
Find the composition function of the functions:
We compute ,
This is valid for all where
. (These are the values for which
.)
Find the composition function of the functions:
We compute ,
This is valid for all .
Find the composition function of the functions:
We compute ,
This is valid for all .
Find the composition function of the functions:
We compute ,
This is valid for all .