An expression like 3*x* + 2 literally stands for a real number;
you simply don't know *which* number
until you know which number the variable *x* stands for.
But when we work with that expression,
we often think about all of the values that it can take, as *x* varies,
at once.
For example, to graph that expression,
we don't just draw a point (for a single value)
or even really a bunch of points;
we draw a *line*.
Geometrically, that line is itself a single complete object in its own right,
something more than a point.
Algebraically, we can also think of an expression
as describing a single complete object in its own right,
something more than a number.
That something is a *function*.

Much as we can write the solution set of an inequality
as, for example, {*x* | *x* < 3},
so we can also write a function
as, for example, (*x* → 3*x* + 2).
However, for historical reasons,
that notation is not used in most algebra books.
However, it is common to give a function a name,
just as we might give the value of an expression a name in a word problem,
and there is special notation for that.
If *f* stands for our function,
then instead of writing
*f* = (*x* → 3*x* + 2),
we write *f*(*x*) = 3*x* + 2,
which means the same thing (if you state it for all values of *x*).
This is convenient notation anyway,
because with it we can calculate, for example,
that *f*(5) = 3(5) + 2 = 17.

(Remember that, while *x* above always stands for a *number*
(even if the value of that number may vary),
*f* here stands for a *function*,
which is not the same type of thing.
So while *x*(5) would mean *x* times 5,
*f*(5) does not work like that.)

Of course, we will also do more of solving equations, graphing expressions, applying algebra in word problems, and everything else that was part of algebra before. We will just be doing this more and more with functions.

Go back to the course homepage.

This web page was written in 2010 by Toby Bartels. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2012w/introduction/`

.