SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (28 page)

Functions are formulas that tell you how to generate one number by using another number.

Functions can be written in a lot of ways. On the SAT, they’ll usually be written in
f
(
x
) notation, also called “function notation.”

Example:

f
(
x
) =
x
3
+ 4   is a function written in function notation.

When we write with function notation, we don’t have to use
f
(
x
) specifically. We could write
g
(
n
),
a
(
b
), or whatever.

Don’t confuse function notation like
f
(
x
) with the multiplicative expression (
f
)(
x
), which means “
f
times
x
”!

When we evaluate a function for a certain number
x
, it means that we plug the number
x
into the function and see what the
f
(
x
) is.

Example:

If our function is
f
(
x
) =
x
3
+ 4 and we want to evaluate the function where
x
= 2, then we get this:

             
              f
(
x
) =
x
3
+ 4

             
              f
(2) = (2)
3
+ 4

             
              f
(2) = 8 + 4

             
              f
(2) = 12

So for our function, when
x
equals 2, the
f
(
x
) equals 12.

The “domain” of a function is the set of numbers on a number line where the function can be evaluated.

Example:

In the function
f
(
x
) =
x
3
+ 4, the domain is all the numbers on the number line, because we can put any value from the number line in for
x
and get a result for
f
(
x
).

In the func
tion f(
x
) = √
x
, the domain is only those numbers that can have a square root. Remember that, on the SAT, you can’t take the square root of a negative number. That means the domain for the function f(
x
) = √
x
is the set of non-negative numbers.

The “range” of a func
tion is the set of numbers that
f
(
x
) can come out equal to.

Example:

The function
f
(
x
) =
x
3
+ 4 has a range of negative infinity to positive infinity—by putting in the right thing for
x
, we can get any number we want as
f
(
x
).

The function
f(
x
) = √
x
has a range of only non-negative numbers, because there is no way to put any number as
x
and get a number for
f
(
x
) that’s negative.

A point can be plotted on a graph in (
x, y
) notation if we take the
x
number and make it the horizontal separation between the point (
x
,
y
) and the origin (0, 0), and then we make the
y
value the vertical separation between (
x, y
) and (0, 0).

A linear function is a function in which the
f
(
x
) is replaced with a
y
, and all the (
x, y
) pairings form a straight line when they’re plotted on a graph.

Example:

The function
f
(
x
) = (
x
/2) + 1 is linear, because all of the (
x, y
) pairings that it generates fall in a straight line when they’re plotted as lines on a graph.

Here’s a chart that shows some (
x, y
) pairings for the function
f
(
x
) = (
x
/2) + 1:

             
                           
x              y

             
                            0              1

             
                            1              1.5

             
                            2              2

             
                            3              2.5

             
                            4              3

             
                            5              3.5

             
                            6              4

             
                            7              4.5

             
                            8              5

             
                            9              5.5

             
                            10              6

             
                            11              6.5

             
                            12              7

             
                            13              7.5

             
                            14              8

             
                            15              8.5

             
                            16              9

When we plot these points on a graph, we see that th
ey fall in a straight line:
 

Of course, the points plotted on the graph are only the (
x, y
) pairings when
x
is a positive integer. But isn’t the domain for
f
(
x
) = (
x
/2) + 1 all the numbers on the number line?  That means that there must be an
f
(
x
) even where
x
equals 1.135623, or 8.4453, or any other number at all. For every value on the
x
axis, there’s a corresponding
f
(
x
) value on the
y
axis. We could “connect the dots” on our graph above, and extend the line of our function infinitely in either direction. Let’s do that:

The “slope” of a linear function is a fraction that shows you how steeply the line is tilted. To find the slope of a line, choose any two points on the line. Measure the vertical separation between the two points,
starting from the left-most point
. The vertical separation number goes in the numerator of the slope fraction. Then measure the horizontal separation between the two points, again starting from the left-most point. The horizontal separation goes in the denominator of the slope fraction.

Exampl
e:

In our graph above, we can
pick any two points on the function line to determine the slope. Let’s pick (2,2) and (8,5). The vertical separation here is the difference between 2 on the left-most point and 5 on the right-most point. So the vertical separation here is 3, and we put a 3 in the numerator of the slope fraction for this line. Now we determine the horizontal separation between 2 on the left and 8 on the right, which is 6. So a 6 goes in the denominator of the slope fraction. Now we have the numerator and the denominator of the slope fraction, and we see that the entire slope fraction is 3/6, or 1/2. So the slope of
f
(
x
) = (
x
/2) + 1 is 1/2. (Keep reading for a much easier way to figure out slope.)

The
equation for a line will often be written in this format:
y
=
mx
+
b
. In fact, our function from the previous example was written in that way:
f
(
x
) = (
x
/2) + 1. (Remember that
y
and
f
(
x
) are the same thing for the purposes of graphing a function, and that “½
x
” can be re-written as
x
/2.)

This
y
=
mx
+
b
format is called “slope-intercept format.”  We call it that because it shows us two things right away: the slope of the function, and the “
y
-intercept” of the function.

The
m
coefficient of the
x
variable will be the slope.

The
b
constant in the function will be the point where the linear function crosses the
y
axis.

Example:

In the linear function
f
(
x
) = 9/7
x
+ 14, the
m
in the
y
=
mx
+
b
notation is 9/7, and the
b
is 14. This means the slope of the function is 9/7, and the point where the line crosses the
y
-axis is 14.

In the linear function
y
= -3/2
x
+ 2, the slope is -3/2 and the
y
-intercept is 2.

When two linear functions have the same slope, they are parallel.

When you can multiply the slope of one linear function by the slope of another linear function and get -1, the two linear functions are perpendicular to one another.