Formulas

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Functions are relations, which makes them packages. They contain three sets. They contain a set of real numbers called the domain. They contain a second set of real numbers called the range. Finally, they contain a third set of pairs of numbers. The pairs are all constructed with a number from each of the domain and range. The defining characteristic of a function is that each domain number is in exactly one pair.

The pairs are the reason we study functions. The pairs are the connection between the domain and range. They identify which numbers are associated together from both sets. They itemize the association between the domain and range.

Therefore, it comes as no surprise, that we have a toolbox full of tools to help us describe these pairs.

Function Notation

We have some written notation to help us communicate about the pairs. $F(d)$ is called function notation. The name of the function is followed by a domain number inside parentheses. $F(d)$ represents the function value at $d$ or the range partner of $d$ . That brings us to at least four uses of parentheses in mathematics.

Parentheses

Mathematics overuses symbols and allows the context in which the symbol is used to affect their meaning.

Grouping: $(3+5)(6+7)$ is a product. Grouping expressions might signal multiplication with the multiplication sign omitted.
Ordered Pairs: $(4, 5)$ is an ordered pair and might represent a function pair.
Coordinates: $(4, 5)$ might represent the coordinates of a point in the Cartesian Plane.
Intervals: $(4, 5)$ might repesent the open interval of real numbers from $4$ to $5$ .
Function Notation: $Double(5)$ symbolizes a range element that is paired with $5$ from the domain in the Double function. It is not multiplication.

The context in which parentheses are used helps the reader interpret the parentheses. And, these contexts can be intertwined.

$((1+4)(7-6), Double(5))$

$F$ , $d$ , and $x$

The actual letter “ $F$ ” itself is not part of function notation. The first part of function notation is the name of the function, which in most instances is not “ $F$ .” The parentheses are always there. The letter $d$ is in the domain position and can be replaced by any expression representing domain elements. We will strive not to overuse $F$ and $d$ , or $x$ .

Graphs

We have visual tools where $F(d)$ is encoded by plotting a dot at $(d, F(d))$ on the Cartesian Plane. $F(d)$ is the signed distance between the point and the horizontal axis. The collection of all dots formed from the function pairs is called the graph of the function. Graphs are drawings, which makes them inherently inaccurate. They are not exact tools. They are global tools. They give a big picture of the function pairs. From this bird’s eye view, we can see trends in the function values, locations of extreme function values, and important function behavior. The global perspective quickly communicates basic characteristics and points us towards function features.

Formulas

We also want exactness!

The price we pay for exactness is that we can only investigate the function, one pair at a time. Our tools for this type of investigation are called formulas and equations.

Video: Formulas are Procedures

[ Click on the arrow to the right to expand for video. ]

Learning Outcomes

In this section, students will

evaluate a function via a formula.
solve equations involving formulas.

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more examples can be found by following this link
More Examples of Formulas