FUNCTION

FUNCTION

A function is a relation (usually an equation) in which no two ordered pairs have the same x-coordinate when graphed. One way to tell if a graph is a function is the vertical line test, which says if it is possible for a vertical line to meet a graph more than once, the graph is not a function. The figure below is an example of a function. Functions are usually denoted by letters such as f or g. If the first coordinate of an ordered pair is represented by x, the second coordinate (the y coordinate) can be represented by f(x). In the figure below, f(1) = -1 and f(3) = 2. When a function is an equation, the domain is the set of numbers that are replacements for x that give a value for f(x) that is on the graph. Sometimes, certain replacements do not work, such as 0 in the following function: f(x) = 4/x (you cannot divide by 0). In that case, the domain is said to be x <> 0. There are a couple of special functions whose graphs you should have memorized because they are sometimes hard to graph. They are the absolute value function (below) and the greatest integer function (below). The greatest integer function, y = [x] is defined as follows: [x] is the greatest integer that is less than or equal to x.
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If nonvertical lines have the same slope but different y-intercepts, they are parallel. 1. Problem: Determine whether the graphs
of y = -3x + 5
and 4y = -12x + 20 are
parallel lines.

Solution: Use the Multiplication Principle
to get the second equation in
slope-intercept form.

y = -3x + 5
y = -3x + 5

The slope-intercept equations
are the same. The two
equations have the same graph.

2. Problem: Determine whether the graphs of
3x - y = -5 and
y - 3x = -2 are
parallel.

Solution: By solving each equation for y,
you get the equations in
slope-intercept form.

y = 3x + 5
y = 3x - 2

The slopes are the same, and
the y-intercepts are different,
so the lines are parallel.
Sometimes, you will be asked to find the equation of a line parallel to another line. Not all the information to put the equation in slope-intercept form will always be given. Example: 3. Problem: Write an equation of the line
parallel to the line 2x + y - 10 = 0
and containing the point (-1, 3).
Solution: First, rewrite the given equation
in slope-intercept form.
y = -2x + 10
This tells us the parallel line
must have a slope of -2.
Plug the given point and the
slope into the slope-intercept
formula to find the y
intercept of the parallel line.
3 = -2(-1) + b
Solve for b.
1 = b
The parallel line's equation is
y = -2x + 1.
If two nonvertical lines have slopes whose product is -1, the lines are perpendicular. Example:
1. Problem: Determine whether the lines
5y = 4x + 10 and 4y = -5x + 4
are perpendicular.
Solution: Find the slope-intercept equations
by solving for y.
y = (4/5)x + 2
y = -(5/4)x + 1
The product of the slopes is -1, so the
lines are perpendicular.
Sometimes, you will be asked to find the equation of a line perpendicular to another line. Not all the information to put the equation in slope-intercept form will always be given. Example:
2. Problem: Write an equation of the line
perpendicular to 4y - x = 20 and
containing the point (2, -3).
Solution: Rewrite the equation in
slope-intercept form.
y = .25x + 5
We know the slope of the
perpendicular line is -4
because .25 * -4 = -1. (Notice
that the slope of the
perpendicular line is the re-
ciprocal of the other line's
slope.)
Now plug the given point and
the slope into a slope-intercept
equation to find the y
intercept.
-3 = (-4)2 + b
Solve for b.
b = 5
Now, you have the information
you need to write an equation
for a line perpendicular to
4y - x = 20. The answer
is the following equation:
y = -4x + 5.