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's a part of Algebra.
In particular, we will look at these
with *exponential*, *logarithmic*, and *trigonometric*
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),
log_{2} *x*, and sin *x*.
(If you don't know what those mean yet, that's OK; we'll cover that!)

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2021 August 23. Toby reserves no legal rights to it.

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