linear linear linear
Given in the domain of , such that is in the domain of , is defined to be .
If and represent the domains of and respectively, the induced or implied domain of is
The order matters. Generally speaking, . Composition is not commutative.
Linear Composition
In this section, we will explore the composition of linear functions.
Let be a linear function with as its domain.
Let be a linear function with as its domain.
Then is a new function. What is its formula?
To keep everything separate, let’s use as the variable for the formula of .
The composition of two linear functions is again a linear function.
Let be a linear function with as its domain.
Let be a linear function with as its domain.
Define two new functions, and , as compositions by
Two different functions. Both are linear functions. The constant rate-of-change is for
both.
Their graphs would be parallel lines.
Let be a linear function with as its domain.
Let be a linear function with as its domain.
Let
Determine a formula for .
Let be a linear function with as its domain.
Let be a linear function with as its domain.
Let
Determine a formula for .
Restricted Domains
Let’s restrict the domains of our linear functions.
Graph of .
Let’s create the composition .
This means the range of needs to be inside the domain of . But as the chart below shows, there are numbers in the range of that are not in the domain of .
For instance, is in the range of , but not in the domain of .
Therefore, we need to remove the numbers in the domain of , whose function value is .
Like, is in the domain of , and . So, has to be removed from the domain of .
What part of the range of can we use?
It looks like we can use the interval . is not included, so we’ll have to keep this in mind as we investigate the domain of .
The range of needs to be restricted to . Therefore we need to restrict the domain of , so that the range is just . We need to find the preimages of and in the domain of .
The preimage of is . The domain of is . is an endpoint of the interval that is already excluded.
Now, for the preimage of .
is inside .
We need to restrict the domain of to
With this restricted domain, the range of will be , which is the biggest piece of the domain of that we can get from values of .
We now have a domain for our composition:
Halfway
Now we need to take our restricted domain of , , and determine the restricted range.
- - included
- - excluded
What do we have?
Our composition has a formula, let’s use for its variable.
It’s domain is .
It’s range is .
Graph of .
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more examples can be found by following this link
More Examples of Piecewise Composition