function on mathematics

BY
ARIJIT BOSE
NILANKO SIKDAR
SWAGATA BISWAS
AKASH DAS
1

FUNCTIONS
A function is a relation in which each element of the
domain is paired with exactly one element of the
range. Another way of saying it is that there is one and
only one output (y) with each input (x)
2

We commonly call functions by letters. Because
function starts with f, it is a commonly used letter to
refer to functions.
This means
Right hand
side has
function
named f.
4

• Variable x is called independent variable
• Variable y is called dependent variable
• For convenience, we use f(x) instead of y.
• The ordered pair in new notation becomes:
(x, y) = (x, f(x))
5

REPRESENTATION OF
FUNCTION• Verbally
• Numerically, i.e. by a table
• Visually, i.e. by a graph
• Algebraically, i.e. by an explicit formula
6

Domain, Co-domain and
Range of a Function
7
F(A) = {f(x)|x∈A} =
A
B

Domain, Co-domain and Range
of a Function
8

TYPES OF FUNCTION
 One-one function (or injective)

9

FOR EXAMPLE :
 One-one function (or injective)
10

TYPES OF FUNCTION
 ONTO FUNCTION
 A function f from a set X to a
set Y is surjective (or onto), or
a surjection, if for every
element y in the co-domain Y
of f there is at least one
element x in the domain X of f
such that f(x) = y.
 FOR EXAMPLE:
11

TYPES OF FUNCTION
˃ Many-one Function : If any two or more elements of set A
are connected with a single element of set B, then we call this
function as Many one function.
˃ Into Function : Function f from set A to set B is Into function if
at least set B has a element which is not connected with any of
the element of set A.
12

INVERSE OF THE FUNCTION
˃ The functions f and g are inverse functions if f(g(x))=x for all x in the domain of g
and g(f(x))=x for all x in the domain of f .
˃ The inverse of a function f is usually denoted and read “ f −1.”
˃ Let f be a function whose domain is the set X, and whose image (range) is the
set Y. Then f is invertible if there exists a function g with domain Y and image X,
with the property: f(x)=y  g(y)=x. A function has to be one-to-one(injective) to
have an inverse. A function has to be bijective in nature.
13

Function Overlapping
 Functions with overlapping domains can be added, subtracted,
multiplied and divided. If f(x) and g(x) are two functions, then for
all x in the domain of both functions the sum, difference, product
and quotient are defined as follows.
14

OPERATIONS ON FUNCTION
Operations:
Sum: (f + g)(x) Quotient:(f/g)(x)
f(x) = 2x+3 and g(x) = x2 f(x) = 2x+3 and g(x) = x2
(f+g)(x) = x2+2x+3 (f/g)(x) = (2x+3)/x2
Difference : (f-g)(x) Product: (f .g)(x)
f(x) = 2x+3 and g(x) = x2 f(x) = 2x+3 and g(x) = x2
(f-g)(x) = (2x+3) − (x2) (f·g)(x) = (2x+3)(x2) = 2x3 + 3x2
15

COMPOSITION OF FUNCTION
Composite Function:
Combining a function within another function. Written as follows: f(g(x)) or
(fog)(x).
The domain fog is set of all numbers
x in the domain of g such that g(x)
is in the domain of f.
We must get both Domains right
(the composed function and
the first function used).
16

17
Given a function g, the composition operator Cg is defined
as that operator which maps functions to functions as Cgf = f
◦ g

COMMON FUNCTION
 Linear Function: Square Function:
f(x) = mx + b f(x) = x2
18

COMMON FUNCTION
 Cubic Function: Absolute Value Function:
f(x) = x
3
f(x) = |x|
19

RELATION
RELATION VS FUNCTION
 A function is a special type of relation in which no input in the
domains cannot have multiple outputs in the co-domain.
SET REPRESENTATION
20
FUNCTIO
N

f(x)=√(x)
 If we take X=4
we will get +2 AND -2 in range set but we know that no
input in the domains cannot have multiple outputs in the
co-domain. we get two values and hence its not a
function
21

f(x)=( x²)
 If we take X=4
We will get just 16
we get one value , hence it is a function.
22

FUNCTIONS IN REAL
WORLD A Thermometer
 Most thermometers come with both
Celsius and Fahrenheit scales.
Students can study a thermometer as
an input/output table. Students often
for their convenience and as per
requirement compare the two scales
and figure out the mystery function
rule -- the formula for converting
one scale to the other.
23

FUNCTIONS IN REAL
WORLD Circumference of a circle
 Circumference of a circle is a
function of its diameter. Students can
measure the diameter and
circumference of several round
containers or lids and record that
data in a table. If diameter is the
input and circumference is the
output, what's the function rule?
 As they divide each container's
circumference by its diameter to find
that rule, they should notice a
constant ratio -- a rough
approximation of pi.
24

FUNCTIONS IN REAL
WORLD Shadows
 The length of a shadow is a function of
its height and the time of day. Shadows
can be used to find the height of large
objects such as trees or buildings. If the
height of the object casting the shadow
you want to measure is known, you can
use a formula to determine the length of
the shadow. Convert the sun’s altitude
from degrees to tangent (written
as“tan(α)”). Calculate the formula L =
(H/tan (α)) to determine the shadow
length.
25

FUNCTIONS IN COMPUTER
SCIENCE Functions are used in different
programming languages to do
specific tasks . There are many
internal functions in respective
languages (for example: max( ) in
C++ which provides the maximum
number from the given inputs).There
are also user defined functions where
we can instruct the way it should
work. There are respective rules for
its implementation in respective
languages.
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function on mathematics

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function on mathematics