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For 3700 years mathematicians and scientists envisioned two quantities varying together but they had no means to represent "varying together". They could represent values of the two quantities, but they did not have a way to represent a relationship between them explicitly.
The advent of the equal sign in the 1500’s changed this. You saw in the previous section how equations can be used to represent relationships between variables both explicitly and implicitly.
Israel Kleiner described the next 300 years, after the use of algebraic notation became common, as mathematicians’ attempt to express relationships between variables in a way they could talk about relationships between variables without knowing a formula that defines the relationship.
For example:
The Moon’s center is always some distance from Earth’s center. More precisely, if we measure time from 00:00, January 1, 00 CE continuously as a number of years and measure the distance between Earth’s and Moon’s centers continuously as a number of kilometers, then at each number of years after 00 CE, the Moon’s center was, or will be, some number of kilometers from the Earth’s center.
How might you represent the relationship between number of years from 00 CE and the distance between Earth and Moon without knowing what that relationship is?
In other words, mathematicians wanted to be able to say things like "the Moon was some specific distance from Earth at any specific number of years", but to say it symbolically.
In 1734, Leonard Euler created our modern notation for representing relationships that are functions, writing,
"If $f\left(\dfrac{x}{a}+c\right)$ denotes any function of $\dfrac{x}{a}+c $…".
Euler’s notation allows us to speak of relationships that are functions without having to say what that relationship is. In the case of our Earth-Moon example, we would say
Let x represent a number of years from 00:00, January 1, 00 CE. Let d be the function that relates values of x to distances between Earth and Moon. Then d(x) represents Moon’s distance from Earth at precisely x years from 00 CE.
Reflection 3.11.1. Convince yourself the relationship given above defines a function. What are the dependent and independent variables? How does it satisfy the criterion for being a function?
Think of the notation d(x), pronounced "dee-of-x", as shorthand for the much longer statement: "d(x) represents the value of the relationship d applied to a value of x".
It is important to recognize "d(x)" is a variable as well as "d(x)" means the value of the relationship d when applied to a value of x.
Figure 3.11.1 captures this line of thinking. Figure 3.11.1a illustrates the conception that each number of years since 00 CE is associated with a number of kilometers between Earth and Moon. Figure 3.11.1b illustrates that if x represents a specific number of years, then $d(x)$ represents the number of kilometers between Earth and Moon at precisely x years.
Figure 3.11.1. (a) Use the letter d to represent a relationship between
variables; (b) Use function notation to represent values of the
variables that are related by the relationship named d. Read "d(x)" as
"The value of the relationship d applied to a value of x," or "The value
of d when d is applied to a value of x."
Many students wonder why we use function notation instead of just using another variable. They wonder why we don’t just say the Moon is y kilometers from Earth x years since the beginning of the year 00 CE.
The answer is without function notation, we cannot represent a distance between Earth and Moon at an arbitrary moment in time. With only the letter y at our disposal, we have no way to represent the distance between Earth and Moon at the moment in time that is 2015.12 years after 00:00, Jan 1, 00 CE (when $x=2015.12$).
Representing this distance at this precise moment in time is easy using function notation. The distance between Earth and Moon when $x=2015.12$ is $$d(2015.12).$$
Suppose we want to represent the variation in the distance between Earth and Moon over the 5-year period from $x=2010.12$ to $x=2015.12$. We can do this easily using function notation by writing
$d(2015.12)-d(2010.12)$
The difference $d(2015.12)-d(2010.12)$ represents the variation in distance between Earth and Moon during the period from 2010.12 years to 2015.12 years after the beginning of 00 CE.
Reflection 3.11.2. Try to represent the statement
"The variation in distance between Earth and Moon during the period from 2010.12 years to 2015.12 years after the beginning of 00 CE"
using just the letters x and y. What difficulties do you encounter?
We could represent the variation in the distance between Earth and Moon over every 5-year period by writing $$d(x)-d(x-5) \text {, or }d(x+5)-d(x)$$
The first expression, $d(x)-d(x-5)$, represents the variation in distance between Earth and Moon during the 5 years before the moment in time represented by a value of x.
The second expression, $d(x+5)-d(x)$, represents the variation in distance between Earth and Moon that happened during the 5 years after the moment in time represented by a value of x.
We could define a new function, D, namely $D(x)=d(x+5)-d(x)$. D(1927.48) then represents the variation in distance between Earth and Moon over the 5-year period from 1927.48 years to 1932.48 years.
D(x) represents the variation in the distance from Earth to Moon over the 5-year period starting at the moment in time represented by any value of x. If $D(x)$ is positive, the moon moved farther from the earth over that time interval. If $D(x)$ is negative, the moon moved closer to the earth over that time interval.
It is worth noting the domain of this function’s independent variable as stated so far is ambiguous. It cannot be all real numbers. Were we to let x be -10 billion, then we would be specifying a moment in time when Earth and Moon did not exist.
Similarly, were we to let x be 10 billion, then we can be sure Earth and Moon would have been absorbed by an expanding Sun.
We should thus specify a domain that covers the period of time we are interested in and which we are confident is meaningful.
We could have also said, for example, that we assume the domain of x is all the years from 00 CE when Earth and Moon exist.
For exercises 1-4:
Samuel’s swimming pool has sloped walls that are rounded at the bottom so the area of the water's top surface varies with the amount of water in the pool. Let p be a function that relates the water’s height in the pool in meters with the pool’s surface area in square meters. Let w represent the water’s height in meters. Then p(w) represents the water’s surface area in square meters at height w.
For exercises 5-11:
Al Unser won the Indianapolis 500-mile race in 1987. Juan Pablo Montoya won the Indianapolis 500-mile race in 2015. Unser completed his race in 3.083089 hours. Montoya completed his race in 3.099033 hours. Imagine they drove their exact races against each other and the race ended when the winner crossed the finish line. Let t represent the number of hours since their imaginary race started.
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