Functions are relations, which makes them packages. We’ll be using our functions to relate measurements, which means they are real-valued functions. Measurements come with units, which we usually have standing ready on the sideline while we investigate the numeric pairs of the function. We just keep in mind that as we begin to use functions to analyze situations, we must consider the units to make sure the puzzle fits together properly.

As relations, functions 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, functions contain a third set of pairs of numbers. The pairs are each constructed with a single number from the domain and a single number from the range.

The pairs are the essence of the function. The pairs are the connections between the domain and range numbers or measurements. They identify which numbers are connected together from both sets. As a function, this set adheres to one rule:

We can communicate about these pairs, one by one, with function notation.

$F$ is the name of the function and $d$ is a domain number. $F(d)$ represents the range partner of $d$. $d$ and $F(d)$ are partnered together in $F$.

When we write down such a pairing, we usually write the domain number on the left and the range partner (or function value) on the right, separated by a comma, and wrapped with parentheses.

Independent of this, we might separately have an alternate expression for the range partner, say $r$. We can communicate this with an equation like

Seeing the Whole Picture

We can certainly analyze a function by talking about individual pairs, however, this is an extremely close-up and zoomed-in perspective. We might also like to talk about the whole collection of pairs all at once. Our first attempt at this is via pictures. We need a way to visually represent a single pair and then convert all pairs to a picture. With this picture, we can analyze the function as a whole, identify important places in the domain, detect trends in the data, and quickly estimate information about the whole function.

These pictures or visual tools are called graphs.

Learning Outcomes

In this section, students will

• encode function pairs into dots.
• decipher dots into function pairs.
• use function notation to communicate about the graph.
• evaluate functions via the graph.
• solve equations involving function notation via the graph.