# Composing functions

When we view graphs as functions, taking a given value of and outputting a particular value of according to a rule , then an interesting new operation arises: composition.

In this step we look at composing functions rather generally, and give examples from the various kinds of linear, quadratic and inverse functions we have already studied.

## Composing functions

If and are functions, then the **composition** is the function defined by:

It means that we first apply to , and then apply to . This is one of those cases where it might be more natural to read right to left.

## Composition of linear functions

Let’s do a few examples involving simple linear functions. If and , then

Note that

and that this is different from .

We can also note that more generally the composition of two linear functions will be another linear function.

Q1(E): If and , find and .

However this is not so simple when we look at compositions of higher degree functions.

## Composition of quadratic functions

If and then

In general we can see that the composition of two quadratic functions is going to be a quartic, or degree four, function.

However the composition of a quadratic function with a linear function will be a quadratic function.

Q2(E): What is the composition of the functions and ?

Q3(M): You buy a new kitchen table. Taxes are and delivery is . If the cost of the table is , then what is your total cost? In this question, there is a bit of an ambiguity, as we are not sure if taxes apply to delivery or not. Suppose we define functions and . Then what are and ? If taxes don’t apply to delivery costs, then which composite represents your total cost?

Q4(C): Compositions are not always defined, so one has to be careful. Let and . For which values of are the compositions and defined?

Q5(C): The morning operations for “brushing teeth” and for “eating breakfast” can be performed as or . Daniel believes that the correct order of operations is because it keeps teeth clean for longer. Norman reckons that breakfast tastes better when we choose the order . What do you think?

## Answers

A1.The compositions are:and

.

A2.The composition is

A3.The compositions are:If taxes don’t apply to delivery costs, then the total cost is given by

A4.The composition can be defined for all positive numbers , sinceHowever, the composition

can be defined only for , since the value of the function is otherwise negative and we cannot take the square root of a negative number.

A5.Let’s take a vote in the discussion.

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