Select all of the subsets upon which is increasing.
relative change
- We are interested in the function values - the range values. But they are
controlled by the domain values.
- We ask questions about the function values, but the answers are in the domain.
Where is the function increasing?
Translation: On which set of domain values do the function values increase?
Where is the function decreasing?
Translation: On which set of domain values do the function values decrease?
A graph provides a global view of all of the pairs, which reveals many of the patterns, features, and characteristics we seek.
However, we must separate the graph from the function. The graph helps us answer the questions, but the graph doesn’t hold the answers. The answers are in the domain and range.
Increasing
Let be a function defined on the domain .
Let be any subset of .
is increasing on provided possesses this property:
For every pair , when then .
There is a gap in . However, for the domain numbers in this set, the associated function values follow the definition of increasing.
The set is another story.
On , we have
- and with
- however,
The function is not increasing on the set , even though it is increasing on and
separately.
Decreasing
Let be a function defined on the domain .
Let be any subset of .
is decreasing on provided possesses this property:
For every pair , when then .
- If a function is increasing on its whole domain, then we just say the function is increasing.
- If a function is decreasing on its whole domain, then we just say the function is decreasing.
This story is a bit misleading. Our story involved . Technically, the equal sign allows
the function to stay steady or constant.
An increasing function could increase in value or stay the same. The points could go
uphill to the right or stay at the same height.
For this reason, we have the word strictly.
Let be a function defined on the domain .
Let be any subset of .
is strictly increasing on provided possesses this property:
For every pair , when then .
Let be a function defined on the domain .
Let be any subset of .
is strictly decreasing on provided possesses this property:
For every pair , when then .
Strictly increasing or decreasing functions cannot keep the same value. They must change as the domain changes.
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more examples can be found by following this link
More Examples of Function Graphs