rates of rates
Functions increase. However, they could increase slowly with a small positive rate of change. They could increase quickly with a large positive rate of change.
Functions decrease. However, they could decrease slowly with a small negative rate of change. They could decrease quickly with a large negative rate of change.
What about compositions?
What happens when you
- compose an increasing function with an increasing function?
- compose an increasing function with an decreasing function?
- compose an decreasing function with an increasing function?
- compose an decreasing function with an decreasing function?
To investigate these we need to remember the definitions of increasing and decreasing.
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 .
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 .
Increasing Increasing
Suppose that both and are increasing functions.
That means that when the numbers going into are increasing, then the numbers
coming out of are increasing.
That means that when the numbers going into are increasing, then the numbers
coming out of are increasing.
Now, consider the composition, .
Suppose the numbers going in are increasing. What are the output numbers doing?
Pretend that the values of are increasing. Then the values of are increasing, since is an increasing function.
These values of , which are increasing, are going into . Therefore, the output of is increasing. But, these are the values of the composition.
When increases, then increases.
Let be an increasing function.
Let be an increasing function.
Consider, .
Suppose and are in the domain of , with .
Then , because is an increasing function.
Then , because is an increasing function.
Increasing Decreasing
Suppose that is an increasing function.
Suppose that is a decreasing function.
That means that when the numbers going into are increasing, then the numbers
coming out of are increasing.
It also means that when the numbers going into are decreasing, then the numbers
coming out of are decreasing.
That means that when the numbers going into are increasing, then the numbers
coming out of are decreasing.
Now, consider the composition, .
Suppose the numbers going in are increasing. What are the output numbers doing?
Pretend that the values of are increasing. Then the values of are decreasing, since is a decreasing function.
These values of , which are decreasing, are going into . Therefore, the output of is decreasing, since is an increasing function. But, these are the values of the composition.
When increases, then decreases.
Let be an increasing function.
Let be a decreasing function.
Consider, .
Suppose and are in the domain of , with .
Then , because is a decreasing function.
Then , because is an increasing function.
Decreasing Increasing
Suppose that is a decreasing function.
Suppose that is an increasing function.
That means that when the numbers going into are increasing, then the numbers
coming out of are decreasing.
It also means that when the numbers going into are decreasing, then the numbers
coming out of are increasing.
That means that when the numbers going into are increasing, then the numbers
coming out of are increasing.
It also means that when the numbers going into are decreasing, then the numbers
coming out of are decreasing.
Now, consider the composition, .
Suppose the numbers going in are increasing. What are the output numbers doing?
Pretend that the values of are increasing. Then the values of are increasing, since is a increasing function.
These values of , which are increasing, are going into . Therefore, the output of is decreasing, since is an decreasing function. But, these are the values of the composition.
When increases, then decreases.
Let be a decreasing function.
Let be an increasing function.
Consider, .
Suppose and are in the domain of , with .
Then , because is an increasing function.
Then , because is an decreasing function.
Decreasing Decreasing
Suppose that both and are decreasing functions.
That means that when the numbers going into are increasing, then the numbers
coming out of are decreasing.
It also means that when the numbers going into are decreasing, then the numbers
coming out of are increasing.
That means that when the numbers going into are increasing, then the numbers
coming out of are decreasing.
It also means that when the numbers going into are decreasing, then the numbers
coming out of are increasing.
Now, consider the composition, .
Suppose the numbers going in are increasing. What are the output numbers doing?
Pretend that the values of are increasing. Then the values of are decreasing, since is an decreasing function.
These values of , which are decreasing, are going into . Therefore, the output of is increasing, because is a decreasing function. But, these are the values of the composition.
When increases, then increases.
Let be a decreasing function.
Let be a decreasing function.
Consider, .
Suppose and are in the domain of , with .
Then , because is a decreasing function.
Then , because is a decreasing function.
Let
Let
Where is the composition, , increasing and decreasing?
We can approximate the critical numbers from the graph.
The critical numbers are approximately , , and .
Algebraically
Now, let’s obtain the critical numbers algebraically.
We have a composition of two quadratic functions. We need to know where they individually increase and decrease.
Both and have a positive leading coefficient. Therefore, they both decrease and the increase. The derivative will reveal the specifics.
This tells us that the critical number is . decreases on and increases on .
This tells us that the critical number is . decreases on and increases on .
Overlaping Intervals
We need to know when the output from crosses , which is where changes its behavior.
We are feeling good, because and , which are the approximations from the graph.
- On this interval is a decreasing function.
- The range of on this interval is .
- On , is an increasing function.
- Therefore, the composition is a decreasing function.
- On this interval is a decreasing function.
- On this interval, .
- On , is a decreasing function.
- Therefore, the composition is an increasing function.
- On this interval is an increasing function.
- On this interval, .
- On , is a decreasing function.
- Therefore, the composition is a decreasing function.
- On this interval is an increasing function.
- On this interval, .
- On , is an increasing function.
- Therefore, the composition is an increasing function.
We now know that
- decreases on
- increases on
- decreases on
- increases on
At there is a local minimum of
At there is a local maximum of
At there is a local minimum of
How do these compare with our graph information?
That is what the graph says!
That is what the graph says!
Wonderful !!!
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More Examples of Composition