You've been
playing with "y
=" sorts of
equations for some time now. And you've seen that the "nice"
equations (straight lines, say, rather than ellipses) are the ones that
you can solve for "y
=" and then
plug into your graphing calculator. These "y
=" equations
are functions. But the question
you are facing at the moment is "Why do I need this function notation,
and how does it work?"

Think back
to when you were in elementary school: Your teacher gave you worksheets
containing statements like "[
] + 2 = 4" and
told you to fill in the box. Now that you're grown, your teacher will
give you worksheets containing statements like "x + 2 = 4" and will
tell you to "solve for x".
Why did your teachers switch from boxes to variables?
Well, think about it: How many shapes would you have to use for formulas
like the one for the area A of a trapezoid with upper base a,
lower base b,
and height h?

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A = (^{h}/_{2} )(a + b)

If you try
to express the above, or something more complicated, using variously-shaped
boxes, you'd quickly run out of shapes. Besides, you know from experience
that "A"
stands for "area", "h"
stands for "height", and "a"
and "b"
stand for the lengths of the parallel top and bottom sides. Heaven only
knows what a square box or a triangular box might stand for. So they switched
from boxes to variables because, while the boxes and the letters mean
the exact same thing (namely, a slot waiting to be filled with a value),
variables are better. Variables are more flexible, easier to read, and
can give you more information.

The same is
true of "y"
and "f(x)"
(pronounced
as "eff-of-eks").
For functions, the two notations mean the exact same thing, but "f(x)"
gives you more flexibility
and more information. You used to say "y = 2x + 3; solve
for y when x = –1".
Now you say "f(x)
= 2x + 3; find f(–1)" (pronounced
as "f-of-x is 2x plus three; find f-of-negative-one").
You do exactly the same thing in either case: you plug in –1 for x,
multiply by 2,
and then add the 3,
simplifying to get a final value of +1.

How do we
go about evaluating functions? First, remember this: While parentheses
have, up until now, always indicated multiplication, the parentheses
do not indicate multiplication
in function notation. The expression "f(x)"
means "plug a value for x into a formula f ";
the expression does not mean "multiply f and x"!
Don't embarrass yourself by pronouncing (or thinking of) "f(x)"
as being "f times x".

In function
notation, the "x"
in "f(x)"
is called "the argument of the function", or just "the
argument". So if they give you "f(2)"
and ask for the "argument", the answer is just "2".

Let's proceed
to "evaluation": You evaluate "f(x)"
just as you would evaluate "y".

Given
f(x) = x^{2} + 2x – 1,
find f(2).

f(2)
= (2)^{2} +2(2) – 1
=
4 + 4 – 1
= 7

Given
f(x) = x^{2} + 2x – 1,
find f(–3).

f(–3)
= (–3)^{2} +2(–3) – 1
=
9 – 6 – 1
= 2

(If you experience
difficulties with negatives,
try using parentheses as I just did above; it helps keep track of things
like whether or not the exponent is on the "minus" sign.) Remember
that "x"
is just a box, waiting for something to be put into it. Don't let odd-looking
problems scare you:

An important
type of function is called a "piecewise"
function, so called because, well, it's in pieces:

As you
can see, this function is split into two halves: the half that comes before x = 1,
and the half that goes from x = 1 to infinity. Which half of the function you use depends on what the value
of x is. If we want to evaluate f(0),
we must, since 0 < 1,
use the first half of the function. Then f(0)
= 2(0)^{2} – 1 = 0 – 1 = –1.
If we want to evaluate f(2),
then we must, since 2
> 1, use the second
half of the function. Then f(2)
= (2) + 4 = 6.
If we want to evaluate f(1),
we must also use the second half, because the first half is only for x's
that are strictly
less than 1.
The second half is for x's
that are greater than or
equal to1.
Then f(1)
= (1) + 4 = 5.

Stapel, Elizabeth.
"Function Notation: Introduction / Evaluating at a Number." Purplemath. Available from http://www.purplemath.com/modules/fcnnot.htm.
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