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,

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Stumbling Robot

A Fraction of a Dot
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Tag: Composition

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Compute values of a composite function given values of the components

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Find points at which a given composite function is continuous

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Find points at which a given composite function is continuous

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Find points at which a given composite function is continuous

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Find a formula for the composition of given functions

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Find a formula for the composition of given functions

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Find a formula for the composition of given functions

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Find a formula for the composition of given functions

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Find a formula for the composition of given functions

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Find a formula for the composition of given functions

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 .