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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 that they could talk about relationships between variables

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.

Think of the notation

It is important to notice that
"d(x)" is a variable as well as that "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

The answer is that without function notation, we cannot represent a distance between Earth and Moon at an arbitrary moment in time. With only the letter

Representing this distance at this precise moment in time is easy using function notation. The distance between Earth and Moon when

$d(2015.12)-d(2010.12)$

The difference of We could represent the change in the distance between Earth and Moon over

We could define a new function,

It is worth noting that the domain of this function’s independent variable as stated so far is ambiguous. It cannot be all real numbers. Were we to let

- Explain why
*p*meets the criterion for a being a function. - Explain the meaning of “
*p*(*w*)”. - Explain the meaning of each statement as it relates to the swimming pool.
- $p(1.8)$
- $p(c) = 14.3$ for some number c
- $p(0.9) - p0.7)$
- $p(1 + a) - p(a)$
- $p(b + 0.5) - p(b)$
- A student in another class claimed that $p(0.9)-p(0.7)=p(0.2)$. Do you agree? Explain.
- Use function notation to represent the distance that each driver traveled in relation to the number of hours since the race began. Be sure to say what every letter you use means. State the domain of your independent variable.
- Represent each of the following using function notation. Explain how what you wrote represents what you claim.
- The distance that Unser traveled in the first 0.09 hours of the race.
- The distance that Montoya travel in the first 97 minutes of the race.
- The distance that Unser traveled from
*t*= 1.192 to t = 2.013 hours. - The distance that Montoya traveled in every one-minute time period.
- Represent the distance between Unser and Montoya when Unser crossed the finish line. Explain how what you wrote represents what you claim.
- Represent the distance between Montoya and Unser at every moment from when the race began to when it ended. Explain how what you wrote represents what you claim.
- Represent those times when Unser was ahead of Montoya. Explain how what you wrote represents what you claim.
- Represent the fact that Unser was ahead of Montoya when Unser finished the race.
- Represent the moment(s) in time when Unser was 0.23 miles ahead of Montoya. Explain how what you wrote represents what you claim. Is it possible that your equation has more than one solution? Why?

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 that they drove their exact races against each other and that the race ended when the winner crossed the finish line. Let