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.
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
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?
A1. The compositions are:
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 , since
However, 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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