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.
Instead, 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
(stating it for all values of *x*),
which means the same thing.
This is convenient notation anyway,
because with it we can calculate, for example,
that *f*(5) = 3(5) + 2 = 17.
(Remember that, while *x* here stands for a *number*,
*f* stands for a *function*,
which is not the same type of thing.
So, 3(5) means 3 times 5, and even *x*(5) would mean *x* times 5,
but *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.
In particular, we will look at these
with *exponential* and *logarithmic* operations;
that is, we will look at expressions like 2^{x}
where *x* is allowed to be *any* real number
(not just a rational number)
and learn how to solve equations involving such expressions.

Go back to the course homepage.

This web page was written between 2010 and 2016 by Toby Bartels, last edited on 2016 March 31. Toby reserves no legal rights to it.

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