$d$ is sitting in the left position of our ordered pair, therefore it represents a domain number. $F(d)$ represents the value of the function at $d$ and is written on the right in the ordered pair.

That was looking at the function $H$ one pair at a time.

In addition to talking about individual pairs, we might also like to talk about the whole collection 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.

Visually Encoding

Let $f$ be a function and $d$ a domain number of $f$.

Our plan is to visually encode the pair $(d,f(d))$ as a dot in the Cartesian plane with coordinates $(d,f(d))$.

From the origin, we measure horizontally a distance of $d$. Then we measure a vertical distance of $f(d)$. Plot a dot. The horizontal and vertical measurements for a point are called its coordinates.

• The horizontal coordinate is the domain number.
• The vertical coordinate is the function value.

Suppose we have a function $M$ and we know that $M(3)=2$. This information can be encoded visually with a dot plotted at $(3, 2)$.

Suppose we have a graph of $y=H(t)$ and one of the following four points is on this graph. Which point would be on the graph if $H(-3) = -2$?

$A$ $B$ $C$ $D$

Note: The horizontal axis is named $t$, because the function notation, $H(t)$, tells us that $t$ represents the domain numbers. We generally name the vertical axis with some name not already in use, like $y$. Then, we let the reader know that the values of $y$ are representing values of the function, $y=H(t)$.

Suppose $k$ is a function with $k(-1)>0$. When we consider the graph of $y=k(t)$, there will be a point corresponding to $k(-1)$. In which quadrant will the corresponding dot be plotted?

$I$ $II$ $II$ $IV$

A graph of a function is just a collection of a bunch of dots. Each dot represents a pair in the function. Each dot has two coordinates. The horizontal (first/left) coordinate is the domain number and the vertical (second/right) coordinate is the function value at that domain value.

The functions we are most interested in have millions of billions of trillions of dots.

As we add more and more dots....

It soon becomes difficult to distinguish the individual dots.

They begin to overlap...

Eventually, your eyes play tricks on you. You think you see a single object. A thing - like a wire bent on a piece of paper.

It isn’t!!!!

It is a bunch of individual dots very close together. They are forming a pattern that your eye likes. Your eye glues them together. Much of this course is about these eye-pleasing patterns that functions create.

Dots

Of course, the dots on a graph of a function are wrong. They are too big. Dots, even the size of a pencil point, cover millions of billions of points. If you doubt that, then look at the graph under a microscope.

Does the graph below tell us that $K(1) = -4$ or $K(1.03) = -4.1$ or $K(0.95) = -3.97$ or $K(1.01) = -3.95$?

The answer is yes.

It says all of those, because the dots are too big. Our graph should consist of points, but points are dimensionless. We wouldn’t be able to see them. So, we draw dots that we can see and accept the inaccuracy of the graph.

Therefore, graphs are inherently inaccurate and there is nothing you can do about that. Graphs are communication tools. We have to keep this in mind when discussing mathematics with other people. Algebra is our tool for exactness. Graphs give us the overall picture, but at the expense of accuracy.

We want to use both Algebra and graphs. They each tell us what the other should be doing. We just have to remember what each tool provides us.

Domain and Range

From our visual encoding plan, we see that the first coordinate of each dot will be a number from the domain of the function and the second coordinate of each dot will be a number from the range of the function. More specifically, the second coordinate is the function value at the first coordinate. The horizontal and vertical axis are seen as holding the domain and range.

• DOMAIN: We think of the domain as sitting on the horizontal axis.
• RANGE: We think of the range as sitting on the vertical axis.

The two axes in our Cartesian plane are performing double duty. We need double vision to see them correctly. If a dot on the graph happens to land on an axis, then we have function value information.

If the graph of $y=K(v)$ included three dots on the axes...

... on one hand...

• The dot at $(-1,0)$ tells us that $K(-1) = 0$.
• The dot at $(0,-3)$ tells us that $K(0) = -3$.
• The dot at $(3,0)$ tells us that $K(3) = 0$.

...on the other hand, from all of the dots, we can see that the domain of $K$ includes $\{ -2, -1, 0, 1, 2, 3 \}$. This is the collection of all of the first coordinates of all of the dots.

What if we want to visualize the domain and the range of the function, by themselves? When visualizing the domain and range, we would view each axes as an individual number line. Rather than thinking of the axes as holding dots with two coordinates, we would think that the axes are number lines, holding numbers, and we would color in the intervals and numbers, like on a number line.

We need our brains to jump back and forth between these views. Dots represent pairs in a function. We decipher these pairs of numbers by the location of the dot. At the same time, locations on the axes also represent domain and range numbers - depending on the plotted dots.

Below is the graph of $y=h(f)$. We can identify its domain, by collecting all of the first coordinates of all of the dots. We can identify its range, by collecting all of the second coordinates of all of the dots.

We could think of the horizontal axis as the domain number line, squeeze all of the dots to the horizontal axis, and draw in the intervals of the domain. $$domain = [-4,0] \cup (1,7]$$

We could think of the vertical axis as the range number line, squeeze all of the points to the vertical axis, and draw in the intervals of the range.

$$range = (-6.5, -3.5] \cup [1, 4]$$

This presents a communication issue.

When we were drawing intervals on numbers lines, we used square brackets and parentheses to signal the inclusion or exclusion of numbers. Now we are using filled and hollow dots.

This is due to the long history of mathematics. We must be ready to read and understand both.

• Filled (or closed) dots and square brackets indicate the inclusion of a number or a point.
• Hollow (or open) dots and parentheses indicate the exclusion of a number or a point.

Unfortunately, the term endpoint is used for the end numbers of intervals, when really they are end-numbers. Again, more language we must be aware of.

In this course, we will try to deliberately and appropriately use number and point. As you discuss mathematics with more people, you will encounter different usage.

Below is the graph of $z=P(k)$. Use it to answer the following questions.

What is the domain of $P(k)$?

$[-5,0] \cup [1,5]$ $[-5,5]$ $(-5,0] \cup [1,5)$ $[-5,0) \cup (1,5]$ $[-5,0), (1,5]$

What is the range of $P(k)$?

$[-6,9]$ $[-6,-2) \cup [0,9]$ $[-6,-2) \cup [0,5)$ $[-6,-2), [0,9]$