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# Compute values of a composite function given values of the components

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,

# Find points at which a given composite function is continuous

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.

# Find points at which a given composite function is continuous

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 .

# Find points at which a given composite function is continuous

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 a formula for the composition of given functions

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 a formula for the composition of given functions

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 a formula for the composition of given functions

Find the composition function of the functions:

We compute ,

This is valid for all where . (These are the values for which .)

# Find a formula for the composition of given functions

Find the composition function of the functions:

We compute ,

This is valid for all .

# Find a formula for the composition of given functions

Find the composition function of the functions:

We compute ,

This is valid for all .

# Find a formula for the composition of given functions

Find the composition function of the functions:

We compute ,

This is valid for all .