measurements
Count, Length, Weight, Volume, Odor, Density, Brightness,
Strength, Pressure, Heat, Temperature, Loudness, Change, Speed,
Direction, Angle, Moisture, Voltage, Current, Tone, Notes,
Satisfaction, Likelihood, Distance, Absorption, Reflection, Position,
Heat, Magnetism, Sweetness, Sour, Focus, Flexibility, Pollution,
Time, Smoothness, Humor, Stress, Area, Rates, Shape, Location,
Orientation, Health, Age...
...it just keeps going and going and going.
We compare all of these to each other:
- Odor vs. Direction
- Pollution vs. Location
- Flexibility vs. Density
- Heat vs. Pressure
- Voltage vs. Shape
- Age vs. Health
Number Functions
We make functions connecting just about everything. In particular, we could make
functions where both the domain and codomain are sets of measurements. These are
the types of functions we study in Calculus. These are the types of functions
we will study in this course. And, since measurements are real numbers
accompanied by a unit, we will frequently temporarily set aside the unit and
analyze functions that connect sets of real numbers with sets of real numbers.
In Calculus, our applied functions will connect sets of measurements with sets of
measurements. However, we usually hold the measurement units off to the side, work
with the numbers, and then bring back the units when we interpret our results. Once
we arrive at any conclusions, then we will interpret our findings within the
context of the situation under investigation and the measurements involved.
Context: The Harpo Chalk Company
The Harpo Chalk company sells chalk in bulk to schools and school districts. In
an effort to increase sales, the company lowers the price per box of chalk
as the order size increases. The price per box is given in the table below.
Table 1. Chalk Prices per Box
For example, if you purchased box of chalk, the first boxes would cost each for a total cost of . The final boxes (boxes numbered to ) would cost each for a total cost of . The total cost of the entire order would be .
We can see from the previous question that more than boxes were ordered. The next
cut-off is at boxes. How much do boxes cost?
- The first boxes cost .
- The next boxes cost .
- The next boxes cost each for a total of .
The total cost for the boxes is . More than boxes were ordered. We need
to buy worth of chalk. These boxes will cost each, which gives us boxes.
Total number of boxes ordered is boxes.
This is called the effective price.
The previous story compares boxes to dollars. Those were the units for the
measurements.
We can also have functions that just relate numbers.
The Double function pairs a real number with its double.
domain = all real numbers
codomain = all real numbers
- Double() = .
- Double() = .
- Double() = .
- .
The Half function pairs a real number with its half.
domain = positive real numbers
codomain = positive real numbers
Solve Half (d) =
The solution is .
The Successor function pairs an integer with one more than the integer.
domain = all integers
codomain = all integers
- Successor() = .
- Successor() = .
- Successor() = .
- Successor() = .
- .
Solve Successor(z) =
The solution is .
Fix a number such that , a real number strictly between and , and let be a function, such that
- the domain of is all real numbers strictly between and
- the codomain of is all real numbers strictly between and
- pairs a domain number with its product with , i.e. .
is well-defined.
To show this, we need only recognize that the product of two real numbers can have
only one value. In addition, the product of two numbers between and is also
between and .
is one-to-one.
To show this, suppose and are two domain numbers and they have the same range
partner: . Since , we have that . Therefore, if two domain numbers have the same
function value, then, in fact, they were not different domain numbers. They were
the same domain number. i.e. a range number cannot be in more than one
pair.
is not onto.
Idea: If you multiply by all of the numbers in , then you get the numbers
.
Now, suppose that is a range number, i.e. a function value. That would mean it is
partnered with some domain number. There would be a domain number , such that .
But that would mean that and , since .
If is in the range, then its domain partner is not in the domain. That can’t happen, which means must not be in the range and is not onto.
The domain and range above were open sets, which gave space between and ,
which allowed for .
Much of our analysis of functions will rest on properties of open sets.
If our function domains and ranges will be sets of real numbers, then it seems we should know about sets of real numbers.
Therefore, we need a way to communicate about sets of real numbers, especially open sets.
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more examples can be found by following this link
More Examples of Functions