UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
INTRODUCTION
In our daily living, we often encounter quantities that do come in pair. For
example, the number of kilograms of rice and the amount of money needed to
purchase. Furthermore, the number of miles a car travelled and the liters of
gasoline consumed. Likewise, the plant growth in centimeters and the amount of
rainfall it received. When one quantity changed, the other also changed. These
pairings are best represented as ordered pairs.
RELATION
If we let the ordered pair be( ), we call a set of ordered pairs as a relation. The
set of all the first elements (the values of x) in the ordered pairs is referred to as
the domain of the relation while the set of all the second elements (the values of
y) forms the range. Thus, in a relation, there is a correspondence between the
domain and range, such that to each element of the domain there is assigned one
or more elements of the range.
The given mapping diagram better
explains the definition of relation, its
domain and range. This relation
consists of five ordered pairs, namely:
( )( )( ) ( )
and( ). Its domain is set
* + and its range is
set* +.
GRAPH OF A RELATION
There is a one-to-one correspondence between the ordered-pairs ( )and
the points on the rectangular or Cartesian plane. Each point on the plane
corresponds to one and only one ordered pair( ). While the domain of a relation
is usually apparent from the definition of the relation, the range is often determined
from its graph. The graph of the above relation consisting of points is shown at the
right.
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
RULE OF CORRESPONDENCE
The rule of correspondence is any equation describing how the elements of
the domain and range of any relation are paired. It virtually gives the range of the
relation. Let us consider three relations described by the same rule of
correspondence but having different domains.
Example 1. Given: Relation *( )| +.
Relation is a set of ordered pairs
consisting of all the possible pairings of the
elements of the domain and range that are
formed according to the given rule of
correspondence. Hence, the elements of
relation are ordered
pairs( ) ( ) ( ). Thus, the domain
of relation is set* +, its range is
set* + and its graph consists of only three
points.
Example 2. Relation *( )| +.
Relation which can simply be denoted as *( )⌋ + consists
of an infinite number of ordered pairs. It is a general rule that if the domain is not
indicated, it means that it consists of all real numbers without any exception. Any
real number that is excluded in the domain must be clearly indicated in the
notation used. This matter is exhibited on the graph of relation which is a line
represented by linear equation y x 2 extending indefinitely up to the right and
down to the left. The domain of relation is * | +, where the real number
set is and its range is* | +. Its graph is shown below.
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
Example 3. Relation *( )| +.
It is understood that the domain of consists of all real values except .
Hence, point ( )is not in set C. This fact is revealed on the graph by drawing
an open circle around the point. Therefore, domain * | +and range * |
+belong to relation .
FUNCTION
Function is a special kind of relation. It is a set of ordered-pairs ( ) of real
numbers in which no two pairs have the same first element . Furthermore, it is a
relation in which each -element has only one -element associated with it.
Relations , and described on the above examples are all functions since for
every value of the first element , there is one and only one corresponding value
of the second element y.
VERTICAL LINE TEST
The vertical line test tells whether a relation is a function. Given the graph
of the relation, if every vertical line drawn crosses the graph in only one point,
then, the relation is a function. On the contrary, if one can draw a vertical line that
goes through two points, is not a function of .
The graph of relations and shown below reveals that is a function
since any vertical line drawn through its graph intersects it in one and only one
point. Moreover, is not a function since any vertical line drawn through its graph
crosses it in more than one point.
Graph of 𝑄 Graph of 𝑊
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
CONSTANT AND VARIABLE
In Mathematics, a constant is a quantity that maintains a fixed value
throughout a particular problem. Absolute constants such as √ retain the
same values in all problems. Arbitrary constants remain constant in a particular
problem but may assume different values in other problems.
A variable is a quantity that may assume various values in the course of a
problem. In equation y 1 x , letter x whose values would be freely assumed is
called the independent variable and letter y whose value depends on the assumed
value of x is called the dependent variable.
FUNCTION NOTATION
To be able to discuss functions and their properties, we use a symbol,
usually a letter of the alphabet to stand for a function. The most often used
are . Sometimes, subscripts are employed so that, for example
and would stand for four different functions. To write a function, we
enclose the independent variable in parentheses preceded by a chosen letter. In
symbol form, ( ), read “function of ”, with the chosen letter indicating that there
exists a relationship between variable x and another variable.
In equation √ , ( ) is read is a function of , with the
Greek letter indicating a relationship between dependent variable and
independent variable , hence, the ordered pair ( )can be denoted by , ( )-
or ( √ ). Function is single-valued function.
Moreover, in function ( ) √ , ( ) is a double-valued function. For
example, when ( ) .
A function that depends on two or more independent variables is
symbolically represented in a similar manner. Hence, a function of variables and
is written as ( ) and is read function of and . The function ( ) when
and is denoted by ( ).
FUNCTION EVALUATION
This is the process of finding value of function, say ( ), given value of the
independent variable . The notation ( ) refers to the value of function
when . Likewise, in ( ), ( ) means the value of the function
when and .
Example 4. Suppose that is a function defined by the equation ( )
. Evaluate ( ) ( ) ( ) ( ) ( ) ( ) , ( )-. Draw the graph of for the
portion of the domain .
Solution: Substituting the given value of the independent variable , we have
f (0) (02 2(0) 3 3
f (1) (1) 2 2(1) 3 1 2 3 0
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
f (2) (2) 2 2(2) 3 4 4 3 5
f (1) (1) 2 2(1) 3 4
f (2) (2) 2 2(2) 3 4 4 3 3
f (3) (3) 2 2(3) 3 9 6 3 0
f f x f x 2 2 x 2 x 2 2 x 3 2 x 2 2 x 3 3
2
x 4 4 x 2 9 4 x3 6 x 2 12 x 2 x 2 4 x 6 3
x 4 4 x3 4 x 2 16 x 12
Tabulating the x values and the corresponding y or f x values,
x 2 1 0 1 2 3
f ( x) y 5 0 3 4 3 0
x, y 2,5 1,0 0,3 1,4 2,3 3,0
The graph of ( ) is a parabola with vertex at ( )
f ( x h) f ( x ) 1
Example 5. Find the value of , h 0 , given function f ( x) 2 .
h x
1
Solution: Evaluate f ( x h)
x h2
x 2 x h
2
1 1
f ( x h) f ( x) x h 2 x 2 x 2 x h 2
Therefore,
h h h
f x h f x x 2 x 2 2hx h 2 h2 x h (h 2 x)
2 2
hx 2 x h hx x h x x h
2 2 2
h
Example 6. Discuss the distinction between the given functions ( )and ( )
9 x2
defined H ( x) , and, G ( x) 3 x
3 x
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
Solution:
At the first glance, it appears that the functions are the same since 9 x 2 is
factorable. However, the domain of G x is x R , meaning, x is any real number.
However, for the function H x , the values of both numerator and denominator are
zero when x 3 . Therefore, ( ) and ( ) are identical for all x -values except
x 3 . The graph of H x has an open circle drawn around the point ( ) since
this point does not lie on its graph.
9− 𝑥 2
Graph of 𝐺(𝑥) 𝑥 Graph of 𝐻(𝑥) +𝑥
𝑥
Example 7: Find the domain and range of function ( x) x 4 .
Solution:
The function ( x) x 4 is defined only at x-values equal or greater than
4. That is, for the function to be a real number, the radicand x 4 0 or x 4
Hence, the domain of the function is x | x 4.
The definition of the given function shows that at values of in the
interval corresponding value of the function is zero or more than zero. That
is same as saying the range of the function is* | +. The graph of function is
the upper half of the parabola with vertex at( ).
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
Example 8. Find the domain and range for the function defined as f ( x) x 2 2 ,
for .
Solution:
The domain of the function f x x 2 2 is x | 2 x 1 . To find the range,
when x 2 , f (2) 6. It could be observed from the graph that range is all real
numbers more than but less than In symbol form, range is y | 2 y 6 . The
graph has an open circle at 2,6 indicating that the domain excludes
and the range does not include .
The graph of the function is a portion of parabola ( ) having
vertex at ( ) opening upward.
Note: If the rule of correspondence defining a given function does not explicitly
point out the domain, one should be sharp enough to identify it. Say for
x
example, f ( x) 2 is a function defined for all values of except ,
x 4
since division by zero is undefined. Similarly, if h( x) 1 x 2 , the domain
consists of values that satisfy the quadratic inequality . Solution
of this inequality and the domain of the function is the interval 1 x 1 . The
graph of the function is the upper half of the circle having center at the origin
and of radius equal to one.
PIECEWISE-DEFINED FUNCTION
This is a function whose domain is divided into parts and each part is
defined by a different function rule. It is defined on a series of intervals. The word
piecewise is used to describe any property of a piecewise-defined function that
holds for each piece but may not hold for the whole domain of the function.
A common piecewise-defined function is the absolute value.
| | {
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
Example 9. Find the domain and range of given piecewise-defined function ( ).
Draw its graph and find value of when and .
( ) 2
Solution: Based on the given parts of the domain, we say that the domain of the
given piecewise-defined function is x | x R. Let us draw the graph of the
given function ( ).
The graph above shows the range of ( ) is * ⌋ + and
when , ( ) . Furthermore, when , ( ) .
Example 10. Find the domain and range of function defined below and draw its
graph.
- x 2 1 , x 1
H ( x) 1 ,x 0
x 2 , x 1
2
The domain of the given function is * | +. Below is
the graph of ( ). Based on the graph of the function ( ) shown below, it is
evident that the range is * | +.
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
Example 11: Find the domain and range of piecewise-defined function ( )
whose graph is shown below. Evaluate H 2 .
Solution: Domain * | + Range is * | +. From the
given graph, the value of the function when is ( ) .
Example 12. Find the domain and range of the function graphed below.
Solution: The domain is * | + while the range is * | +.
Example 13. Find the domain and range of y 25 x 2 .
Solution: The graph of the function is the lower half of circle having
its center at the origin ( ) and radius equal to 5. For the value of to be
real, . This inequality has solution 5 x 5 . Hence, the domain of the
function is * | +. The value are excluded on the
domain as indicated by the open circles at those values of . And from its graph
below, it is evident that the range of the function is * | +.
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
SAQ1
ACTIVITY 1.1 – A
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Identify which of the following representations is/are a function and not a function. Write F if a
function and NF if not a function on the space provided before each number.
__________ 1. *( ) ( )( )( )+
__________ 2. *( ) ( )( )( )( )( )+
__________ 3. *( )( )( )( )+
__________ 4. {*( )| +}
__________ 5. {*( )| +}
__________ 6. 2{( )| √ }3
__________ 7. {2( )| 3}
+
__________ 8. {2( )| 2− 3}
__________ 9. {*( )| +}
__________ 10. 2{( )| √ }3
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ASAQ1
ACTIVITY 1.1 – A
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Identify which of the following representations is/are a function and not a ANSWER
function. Write F if a function and NF if not a function on the space provided
before each number.
__________ 1. *( ) ( )( )( )+ F
__________ 2. *( ) ( )( )( )( )( )+ F
__________ 3. *( )( )( )( )+ NF
__________ 4. {*( )| +} F
__________ 5. {*( )| +} F
F
__________ 6. 2{( )| √ }3
F
__________ 7. {2( )| 3}
F
+
__________ 8. {2( )| 2− 3} F
__________ 9. {*( )| +} NF
NF
__________ 10. 2{( )| √ }3
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
SAQ2
ACTIVITY 1.1 – B
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Given the graph of a relation, determine its domain and range. Write answer on the space
provided under the given graph.
1. 2.
Domain:______________________________ Domain:____________________________
Range:________________________________Range: ______________________________
3 4.
Domain:______________________________Domain: ____________________________
Range: _______________________________Range: ______________________________
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ASAQ2
ACTIVITY 1.1 – B
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Given the graph of a relation, determine its domain and range.
1. 2.
Domain: * | + Domain: : * | +
Range: * | + Range: * | +
3 4.
Domain: * | + Domain: * | +
Range: * | + Range: * | +
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ACTIVITY 1.1 – C
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Given the following relations, identify the domain and range and draw their graphs.
1. A 3,6 , 0,3, 2,1, 4,1
2. B x, y y 2 x 4
3. C x, y y 2 x 4, x 1
1
4. D x, y y
x
5. E x, y y x3
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
6. *( )| +
7. G x, y y 3 2 x
8. H x, y y 4 x 1
9. I x, y x 2 y 2 4
10. J x, y y 24 2 x x 2
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ACTIVITY 1.1 – D
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Given the piecewise-defined functions, draw the graph and identify the domain and range.
1. ( ) 2
2. ( ) {
3. ( ) {
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ACTIVITY 1.1 – E
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Given the graph of piecewise-defined functions, determine its domain and range.
1. 2.
Domain: _______________________________ Domain: ________________________________
Range: _________________________________ Range: _________________________________
3. 4.
Domain: _______________________________ Domain: ________________________________
Range: _________________________________ Range: _________________________________
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ACTIVITY 1.1 – F
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Evaluate the given functions at the indicated values of .
1. Given: f ( x) 2 x , find:
2
a. f (3) = _____ c. f 1 = _____ e. f 1 = _____
b. f 2 = _____ d. f 0 = _______ f. f 2 = _____
Draw the graph of f (x) for 3 x 2 .
2. Given: g ( x) x 2 x 1 , find:
2
a. g (4) = _____ c. g 2 = _____e. g 0 = _____ g. g 2 = _____ i. g 4 = _______
b. g 3 = _____ d. g 1 = _____ f. g 1 = _____ h. g 3 = _____ j. g (a 1) = ____
Draw the graph of g (x) for 4 x 4 .
3x 4
3. Given: ( x) , find:
2x 3
a. (4) = _____ d. 1 = _____ g 2 = _____
b. 3 = _____ e. 0 = _____ h. 3 = _____
c. 2 = _____ f. 1 = _____ i. 4 = _____
Which value of x is not an element of the domain? Draw the graph of x for x on 4,4 using
the values above and additional values, if needed.
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ACTIVITY 1.1 – G
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
x 1 3
1. Given the function g ( x) x and h( x) , find hg (x) and g h( ) .
2
1 x 2
2. Given: f ( x) x x 4 , find , (
2
)-.
x 1
3. Given the function g ( x) x and h( x)
2
, find , ( )- and 0 . /1.
1 x
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UNIT 1-FUNCTION AND LIMIT OF A FUNCTION
ACTIVITY 1.1 – H
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
2x2
A. Prove: If f x , show that f ( x) f ( x) .
x 4 16
B. If r ( x)
( )− ( )
x , show that − ( )+ ( )
.
C. Given h( x) x 2 4 x 5 , what is the domain of function h ? Plot the graph of h for x values
in the interval 2,6 .
Differential Calculus Module 1-Relation and Function Page 20
