Applied Calculus New Free Lecture 2.1 .ppt

APPLIED CALCULUS
Lecture # 02
Introduction to Functions
1

Fair Use Notice
The material used in this presentation i.e., pictures/graphs/text,
etc. is solely intended for educational/teaching purpose, offered
free of cost to the students for use under special circumstances of
Online Education due to COVID-19 Lockdown situation and may
include copyrighted material - the use of which may not have been
specifically authorized by Copyright Owners. It’s application
constitutes Fair Use of any such copyrighted material as provided in
globally accepted law of many countries. The contents of
presentations are intended only for the attendees of the class being
conducted by the presenter.
2

INTRODUCTION TO FUNCTIONS
The term function was recognized by a
German Mathematician Wilhelm LEIBNIZ in
1673.
Why? To describe the dependence of one
quantity to an other quantity. For example
1. Area of circle depends on radius of circle
2. No: of shirts depends on revenue
3. Age depends on height
4. Arm length is a function of height
5. A weekly salary is a function of the hourly pay rate and
the number of hours worked.
20 EL 3

Functions
1. Function is input and out put device.
2.A function is like machine that assigns a
unique output to every allowable input.
EL 20 4

Functions
3. A function is a relation that gives a single output
number for every valid input number (x values
cannot be repeated)
4.Let A and B be nonempty sets. A function f from
A to B is an assignment of exactly one element of B
to each element of A We write f(a)=b if b is the
unique element of B assigned by the function f to
the element a of A.
Functions are sometimes called mappings or
transformations
EL 20 5

Functions
5.A function f from a set X to a set Y is an assignment
of exactly one element of Y to each element of X.
We write f(x) = y or y = f(x)
if y is the unique element of B assigned by the
function f to the element x of A.
If f is a function from A to B, we write f: AB
6.Function is rule to which assigns a value of independent
variable which corresponds to unique value of dependent
variable.
20 EL
6

Functions
If f:AB, we say that A is the domain of f and B is the
co domain of f.
If f(x) = y, we say that y is the image of x and x is the pre-image
of y. The range of f:AB is the set of all images of elements of
A.
We say that f:AB maps A to B.
20 EL 7

Function
7.A function is a rule that maps a number to another
unique number.The input to the function is called the
independent variable, and is also called the argument
of the function. The output of the function is called
the dependent variable.
A Swiss mathematician Leon-Hard Euler invented a
symbolic way to write statement y is function of x as
y = f(x) read as y is equal to f of x where y is called
dependent and x is called independent variable
Example:
y = x + 1
9

Functions
Function: for every x there is exactly one y.
• Domain= the set of x values
• Range = the set of y values
10

Representation of Function
EL 20 11

EL 20 12

EL 20 13

Types of Functions
•A function f:AB is said to be one-to-one (or injective),
if and only if
•x, y  A (f(x) = f(y)  x = y)
•In other words: f is one-to-one if and only if it does not
map two distinct elements of A onto the same element of
B. or
•Distinct elements of A have distinct images
•Different pre images have different images
14

Types of Functions
•Example:
•f(Ali) = Sukkur
•f(Munir) = Karachi
•f(Nek) = Hyderabad
•f(Kaleem) = Karachi
•Is f one-to-one?
•No, Muneer and Kaleem
are mapped onto the
same element of the
image.
15
g(Ali) = Sukkur
g(Munir) = Karachi
g(Nek) = Hyderabad
g(Kaleem) =Rohri
Is g one-to-one?
Yes, each element is
assigned a unique
element of the image.

Types of Functions
•How can we prove that a function f is one-to-one?
•Whenever you want to prove something, first take a look
at the relevant definition(s):
x, yA (f(x) = f(y)  x = y)
•Example:
f:RR
f(x) = x2
Disproof by counterexample:
f(3) = f(-3), but 3  -3, so f is not one-to-one.
16

Types of Functions
•A function f:AB is called onto, or surjective, iff for every
element y  B there is an element x  A with f(x) = y
f is onto if and only if its range is its entire co domain. e.g.
•A function f: AB is a one-to-one correspondence, or a
bijection, if and only if it is both one-to-one and onto.
17
3
y x


Inversion
•An interesting property of bijection is that they have
an inverse function.
•The inverse function of the bijection f:AB is the
function f-1:BA with
•f-1(y) = x whenever f(x) = y.
19

Inversion
20
Example:
f(Ali) = Karachi
f(Ahmed) = Hyderabad
f(Kamran) = N.Shah
f(Parvez) = Sukkur
f(Haleem) = Larkana
Clearly, f is bijective.
The inverse function f-1
is given by:
f-1(Karachi) = Ali
f-1(Hyderabad) = Ahmed
f-1(N.shah) = Kamran
f-1(Sukkur) = Parveez
f-1(Larkana) = Haleem
Inversion is only possible
for bijections(= invertible
functions)

Types of Function
• Constant Function:
Let A and B be any two nonempty sets, then a
function f from A to B is called Constant Function
if and only if range of f is a singleton.
• If f is constant then f(x) = C
• Algebraic Function: The function defined by
algebraic expression are called algebraic function.
21

Functions
• an is called the leading coefficient
• n is the degree of the polynomial
• a0 is called the constant term
Polynomial Function
A polynomial function of degree n in the variable x is
a function defined by
where each ai is real, an  0, and n is a whole number.
0
1
1
1
)
( a
x
a
x
a
x
a
x
P n
n
n
n 



 
 

Polynomial Functions
Polynomial
Function in
General Form
Degree
Name of
Function
1 Linear
2 Quadratic
3 Cubic
4 Quartic
The largest exponent within the polynomial determines
the degree of the polynomial.
e
dx
cx
bx
ax
y 



 2
3
4
d
cx
bx
ax
y 


 2
3
c
bx
ax
y 

 2
b
ax
y 


Even and Odd Functions
A function is y = f(x) is even if, for each x in the
domain of f, f(-x) = f(x)
A function is y = f(x) is odd if, for each x in the
domain of f,
f(-x) = -f(x)
An even function is symmetric about the y-axis.
An odd function is symmetric about the origin.

Ex. g(x) = x3 - x
g(-x) = (-x)3 – (-x) = -x3 + x =
-(x3 – x)
Therefore, g(x) is odd because f(-x) = -f(x)
Ex. h(x) = x2 + 1
h(-x) = (-x)2 + 1 = x2 + 1
h(x) is even because f(-x) = f(x)

Composition
•The composition of two functions g:AB and
f:BC, denoted by fg, is defined by
•(fg)(a) = f(g(a))
•This means that
• first, function g is applied to element aA,
mapping it onto an element of B,
• then, function f is applied to this element of
B, mapping it onto an element of C.
• Therefore, the composite function maps
from A to C.
26

Composition
•Example:
•f(x) = 7x – 4, g(x) = 3x,
•f:RR, g:RR
•(fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101
•(fg)(x) = f(g(x)) = f(3x) = 21x - 4
27

Composition
•Composition of a function and its inverse:
•(f-1f)(x) = f-1(f(x)) = x
•The composition of a function and its inverse is the
identity function i(x) = x.
28

Square root function
•Composition of a function and its inverse:
•(f-1f)(x) = f-1(f(x)) = x
•The composition of a function and its inverse is the
identity function i(x) = x.
29

Floor and Ceiling Functions
•The floor and ceiling functions map the real numbers
onto the integers (RZ).
•The floor function assigns to rR the largest zZ with z
 r, denoted by r.it is also called greatest integer.
•Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4
•The ceiling function assigns to rR the smallest zZ
with z  r, denoted by r.it is also called least integer.
•Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3
30

Applied Calculus New Free Lecture 2.1 .ppt

More Related Content

Similar to Applied Calculus New Free Lecture 2.1 .ppt (20)

Recently uploaded (20)

Applied Calculus New Free Lecture 2.1 .ppt