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TITLE: FUNCTIONS AND RELATIONS
Learning Competencies:

1
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2
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3
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4
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A function is a relationship between two set of variables, x and y, such that each value of set x is associated
with exactly one value of set y, the first set of independent variables is the domain, while the second set
of dependent variables is the range
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Therefore, for each value of x, there corresponds exactly one value of y
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For instance, the amount of money
a worker makes is a function of how many hours he works
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When an independent variable corresponds to more than one variable, it is a relation
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Hence, a relation is a correspondence between a first set of
variables and a second set of variables such that for some elements of the first set of variables, there
correspond at least two elements of the second set of the variable
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There are procedures to show
that a set of ordered pairs, an equation or a graph represents a function
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Otherwise, the relationship is just a mere relation
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x = {Joe, Paul, Romy, Tom} y = {80, 78, 95, 85}
If each of the students in set x corresponds with his grade in set y, the result is a set of ordered pairs of
the form (x, y), as follows:
{(Joe, 80), (Paul, 78), (Romy, 95), (Tom, 85)}

a

Since each student in set x corresponds with exactly one grade in set y, then the set of ordered pairs is a
function
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Example 3: If g = {(2, 3), (5, 6), (7, 8), (2, 4)}, then g is not a function because when
x = 2, there are two different possible values for y, 3 and 4
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We need to demonstrate that for all possible values of
x, there corresponds a unique value of y or f(x)
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Solution: Assign some values for x and compute for the corresponding values of f(x)
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The correspondence is many-to-one
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Therefore,

the equation

Example 6: Determine if ( )

√

is a function
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x
f(x)

1
1

The equation

4
2

9
3

16
4

represents a function
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A relation is a function if the
vertical line intersects or passes through its graph in only one point
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The following relations are examples of functions
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The following examples showing the vertical line intersecting or passing through the graph of each of
the relations in two or more points are not function
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And since the absolute value of any real number is always positive,

then the range is composed of the set of positive real numbers
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To determine the domain and the range of a function, observe the following guidelines:

1
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Thus, we shall define
only for a non-negative real number where the domain is x 0
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2
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In such a case the domain is the set of all real numbers
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Thus, the range in this case is also the set of real numbers
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The function
is the same as
the set of non-zero real numbers where x 0
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For the function
real numbers where y 0
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Hence the domain and the range are composed of

, the domain is the set of real numbers and the range is also the set of
, the domain is x

x in terms of y
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is defined x - 2

0
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And since x always exists for any real number assigned to y, then the

Find the domain and range of the following functions
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Example 2:
The domain is the set of real numbers and the range is 8 since there is only one element
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Example 4:
Both domain and range are made up of the set of real numbers
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Example 5:
Since the square root of a negative number is imaginary/undefined, we cannot substitute values
for x to x + 2 that results to a negative value
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Hence, by the law of inequalities, the value of x in x + 2 0 is x -2which is, therefore, the

domain of the function
to y
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The range is y

0 because any non-negative value can be assigned

Example 6:
The domain of the function is the set of real numbers, excluding 1 because if

x = 1, the value

of y is which makes the function undefined
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