There are many different types of mathematical objects that we could study in this class. Some of them are relation-like objects:

- truth values,
- sets,
- relations,
- ternary relations,
- etc;

- constants,
- functions,
- binary functions,
- ternary functions,
- etc.

That is, a truth value requires 0 dimensions to graph, a constant or a set requires 1 dimension to graph, a function or a relation requires 2 dimensions to graph, a binary function or a ternary relation requires 3 dimensions to graph, and so on; and a truth value or a constant requires 0 variables to define, a set or a function requires 1 variable to define, a relation or a binary function requires 2 variables to define, a ternary relation or a ternary function requires 3 variables to define, and so on.

A **truth value** is either true or false;
any statement with no variables in it,
such as the statement that 0 < 2,
should evaluate to true or false (in this case, true).
To indicate that you are talking about the truth value of this statement,
rather than asserting the statement itself,
you can put curly braces around it
(although there are several other notations used for this);
for example, {0 < 2} is
the truth value that 0 is less than 2,
which is the true truth value rather than the false one.
The graph of the true truth value is a solid dot,
while the graph of the false truth value is a hollow circle;
either way, this takes zero dimensions.
You can also use a variable to give a name to a truth value,
so maybe *p* stands for {0 < 2};
we won't need to do that in this course,
but you'll do it constantly if you take a course on Logic.

A **constant** is, in this class, usually a *real number*,
such as −2.
Any expression with no variables should evaluate to a constant
(possibly undefined),
and we use one dimension to graph a constant on a number line.
Again, you can use a variable to stand for a constant,
so maybe *a* stands for −2;
in other words, *a* = −2.

A **set** is, in the simplest case,
a *set of real numbers*.
A statement with one variable defines a set,
such as {*x* | *x* < 2},
the set of real numbers that are less than 2.
We again use one dimension to graph a set.
If *A* stands for the set
{*x* | *x* < 2},
then these two statements mean the same thing:

*x*∈*A*, usually pronounced ‘*x*in*A*’;*x*< 2.

A **function**, or *unary function* for emphasis,
is a rule
for taking a number (the *input*)
and using it to calculate a number (the *output*).
An example is (*x* ↦ *x* − 2),
the rule which subtracts 2 from any number.
To graph a function, we need two dimensions,
one for the input and one for the output.
If *f* stands for the function
(*x* ↦ *x* − 2),
then these two expressions mean the same thing:

*f*(*x*), usually pronounced ‘*f*of*x*’;*x*− 2.

A **relation**, or *binary relation* for emphasis,
is a set of ordered pairs instead of a set of individual numbers.
An example
is
{*x*, *y* | *x* + *y* < 2}.
We again use two dimensions to graph a relation.

We can continue with binary functions, ternary functions, etc, which take two or more numbers as inputs; and we can continue with ternary relations, quaternary relations, etc, which relate three or more numbers. But we will not actually study these in this class.

One of the basic principles of the theory of functions is that the only information necessary to specify a function is to show how to calculate its value at any argument. Therefore, if I write

and state that this holds forf(x) =x− 2

Or we can say thatf(5) = (5) − 2 = 3.

Notice that I always put parentheses around an expression when I substitute it for a variable; in this case, it wasn't really necessary, but it's best to play it safe.f(2x+ 3) = (2x+ 3) − 2 = 2x+ 1.

Every function can be thought of as a relation, specifically

- {
*x*,*y*|*y*=*f*(*x*)}.

- A graph of a relation is the graph of a function if and only if every vertical line goes through the graph at most once.

Sometimes a vertical line doesn't go through the graph at all!
This happens when *f*(*x*) is undefined.
The **domain** of *f*
is the set of all inputs where *f* is defined:

- dom
*f*= {*x*|*f*(*x*) exists}.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2024 August 15. Toby reserves no legal rights to it. The diagram was drawn with the aid of Jacques Distler's SVG editor.

The permanent URI of this web page
is
`https://tobybartels.name/MATH-1150/2024FA/functions/`

.