same characteristics
How do we describe a shifting domain?
We want a graphical description. We want an algebraic description.
Below is the piecewise-defined function, . The variable is representing the domain values from the set .
On the interval , the graph is a line segment for , a linear function with a restricted
domain.
On the interval , the graph is another line segment for , a linear function with a
restricted domain.
Graph of .
The domain of has two maximal intervals:, and . These correspond to two line segments on the graph. The endpoints give us four strategic points on the graph:
- , which is an open dot on the graph.
- , which is a closed dot on the graph.
- , which is a closed dot on the graph.
- , which is an open dot on the graph.
A New Function
is a new function we are creating, but its definition is based on .
Define by with the induced domain.
First Question: What is the domain of ?
is representing the values of the domain of and is now representing the domain values of . Therefore must take on the values in .
must be in a similar set, but shifted to the left by , because we need to subtract from to get .
The domain of is . It was induced from the domain of through the definition of , which is using . The values in the domain of are real numbers, , such that is in the domain of .
The domain has shifted.
But, the structure of the function hasn’t changed. The same formulas are in use. They are just used by different domain numbers.
Second Question: The graph of had four strategic points. What are the corresponding points on the graph of ?
We have four strategic points on the graph of :
- , which is an open dot on the graph of .
- , which is a closed dot on the graph of .
- , which is a closed dot on the graph of .
- , which is an open dot on the graph of .
How do these translate to strategic points on the graph of ?
- tells us that . To match the template , we need . Replacing with gives us and the point is on the graph of . It is an open dot.
- tells us that . To match the template , we need . Replacing with gives us and the point is on the graph of . It is a closed dot.
- tells us that . To match the template , we need . Replacing with gives us and the point is on the graph of . It is a closed dot.
- tells us that . To match the template , we need . Replacing with gives us and the point is on the graph of . It is an open dot.
Third Question: What is the formula for ?
Graph of .
When we look at the graphs side-by-side, we can see that the graph of has simply shifted left to become the graph of .
From the definition, , we can see that , or . All of the -values are less than the -values. is shifted left .
Notice that four different variables, , , , and , are used to describe the domain and range values of and . We are using separate names for the different types of values to keep everything clear. If we use the same variable to represent both domains, then we run into problems.
For instance... Suppose we use to represent values in both domains. Then is a possibility for . . However, if we write and is also representing the domain of , then we are implying that is a valid number. However, is not in the domain of .
We use variables to help us think about structure and relationships. For this reason, we should not use the same variable to represent values from two different sets in the same situation.
Similarly, the two vertical axes have been labelled and . If we use to represent both axes, then we are using to represent both ranges and that also leads to communication problems.
We will avoid using the same variables or labels for values from different sets.
A graph always helps our thinking. Here is the graph of .
Define a new function by with the induced domain.
The domain of is . These values are represented by in this definition, because is the expression inside the parentheses for and that is where the expression lives for the domain values of .
represents the domain values of . If , then . This is the domain for . It is shifted to the right from the domain of . The individual maximal intervals become , , and .
The graph has the same three shapes. The corresponding endpoints are hollow or filled as they were in the graph of .
Let be a function with domain .
Let be defined as a shifted domain version of . The domain of is .
- In terms of ,
- In terms of , .
When a new function is defined as a domain shift of an existing function, then there
are two different domains.
It is natural to think, ”What was done to the old domain to get the new one?”
Suppose is an existing function with domain .
Define a new function as with the induced domain.
- “” represents elements of the old domain of .
- “” represents elements of the new domain of .
It is natural to ask “what happens to to get ?”
But that is NOT how it is presented to us.
We are told that . But that is what happens to to get . That is backwards of how we naturally think about old and new. We need to solve for to see what is done to .
To get the new domain number, , add to the old domain numbers . This tells us that the graph of will be shifted to the right by , compared to the position of the graph of .
It is reversed from how it was originally defined, , because the “inside” of is here. That is not what is done to . When we solve for , we reverse all of the arithmetic and discover what was done to .
Let be a function with its domain.
Let be defined as with its induced domain.
Then the domain of is the domain of , but shifted Left Right by 5 6 7
Then the graph of is the graph of , but shifted Left Right by 5 6 7
Let be a function with its domain.
Let be defined as with its induced domain.
Then the domain of is the domain of , but shifted Left Right by 1 2 3
Then the graph of is the graph of , but shifted Left Right by 1 2 3
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more examples can be found by following this link
More Examples of Shifting