UNIT II. A Theory of Relations and Functions
In the previous unit we discussed properties of numbers, and how the properties are used to evaluate
expressions and solve equations.
In this unit we will concentrate on equations that have two variables.
These
equations tell how one variable is
related
to the other.
Consider the equation A= πr
2
, where A is the area of the
circle and r is the radius.
For any r, we can find a value for A, say if r
= 1then A = π, if r=2 then A = 4π and so on
depending on the value of the radius.
OBJECTIVES:
At the end of this unit, you must be able to:
1.
define relations, functions and inverse functions
2.
state
the domain, range, intercepts, and symmetry of the functions and relations
3.
differentiate relations from functions
4.
perform operations on functions and
5.
sketch the graphs of
functions and their inverses.
2.1 Relations and Graphs
Recall the definition of the cross product:
Let
A
and
B
be non empty sets.
The
cross product
of A and B
, denoted by A x B is the set of all ordered
pairs (
x
,
y
) such that
x
A
and
y
B
.
In symbols,
A x B = {(
x
,
y
)
x
A and
y
B}.
The ordered pairs
are also called
points
of the cross product.
For example, if A = {1, 2, 3} and B = {a, b} then A x B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}. The graph of
A x B is given below:
Note that if
A
and
B
are both equal to
R
, the set of real numbers, then
A x B = R
2
= {(
x
,
y
)
x
R and
y
R}.
This is exactly the
Cartesian pla
ne or the
coordinate plane
.
The plane can be obtained by drawing two
perpendicular real number lines that intersect at the origin O.
The horizontal line is the
xaxis
and the vertical
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line, the
yaxis
.
In a coordinate plane, each point represents uniquely an ordered pair of real numbers (x, y) and
each ordered pair of real numbers is represented by a point. If the point P = (x, y) then P is x units away from the
yaxis and x units away from the xaxis. We say x is the abscissa and y is the ordinate of P. Taken together, x and
y are called coordinates of P. Thus, if P = (3, 2) then P is 3 units to the right of the yaxis and 2 units above the x
axis. Our convention is that if x > 0, the point is x units to the right of the yaxis and if x < 0, the point is
–
(x)
units to the left of the yaxis. Similarly, if y > 0, the point is y units above the xaxis and if y < 0, the point is
–
(y)
units below the xaxis.
The axes divide the plane into four quadrants as can be seen in the figure below.
TIME TO THINK
1)
Define each quadrant and axis in terms of the characteristics of the points that compose it.
2)
Define distance between two points on a vertical line; on a horizontal line; in the plane.
3)
Draw the straight line joining P
1
(2, 3) and P
2
(3, 2).
4)
Define slope of a line. Find the slope of the line in #3. Describe the line whose slope is 0; positive;
negative.
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 Spring '11
 dikopaalam
 Math, Equations, DOM

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