FUNCTIONS
CLASSIFICATION
OF FUNCTIONS
Function can be classified mainly into two groups
1.Algebraic
Functions
2.Transcendental
Functions
1. Algebraic Functions
A
function which consists of a finite number of terms involving integral and/or
fractional powers of the independent variable (argument) x, connected by the four operators +, -, × and ÷ is called an
algebraic function.
Three particular cases of algebraic functions are the following:
i)Polynomial
Function
A
function of the form a0xn
+ a1xn-1 + …… + an where n
is a positive integer and a0,
a1, …… an are
real constants and a0 ≠ 0 is
called a polynomial in x of degree n.
Example:
F(x) = 2x4
- 5x-3 – x2+ 9x - 6
ii)Rational
Function
A
function of the form f(x)/g(x), where
f(x) and g(x) ≠ 0 are polynomials is called a rational function.
Example:
F(x) = (3x-3
– 9x - 6) / (3x+1)
iii)Irrational
Function
A
function involving radicals, viz., fractional powers of polynomials is called
an irrational function.
Example:
F(x) = (√3x-3
– 9x - 6) / (√3x+1)
2.
Transcendental Functions
A
function which is not algebraic is called a transcendental function.
Example:
i.
Inverse circular functions
ii.
Logarithmic functions
iii.
Hyperbolic functions
iv.
Exponential functions
i. Identity
Function
A
function f : A → A, where f(x) = x, where x ϵ A is called the identify function on A.
In
other words, the identity function is the function that assigns each element of
A to itself and is denoted by IA or simply I.
The function IA is
a bijection.
ii.Floor
Function
If
x is a real number, the function that assigns the largest integer
that is less than or equal to x is called the floor function of x or
simply the floor of x and denoted by ⌊x⌋.
The floor of
x is also called the greatest integer function.
If ⌊x⌋ = n, where n is an integer, then n≤ x< n+1.
Example
⌊6.8⌋ = 6
⌊-6.8⌋ = -7
⌊6.234⌋ = 6
⌊5⌋ = 5
⌊-3⌋ = -3
iii.Ceiling
Function
If
x is a real number, the function that assigns the smallest integer
that is greater than or equal to x is called the ceiling function
of x or simply the ceiling of x and denoted by ⌈x⌉.
If ⌈x⌉ = n,
where n is an integer, then n-1≤ x<n.
Example
⌈6.8⌉ = 7
⌈-6.8⌉ = -6
⌈6.235⌉ = 7
⌈8⌉ = 8
⌈-9⌉ = -9
Note:
It is obvious
that if x is itself an integer, then ⌈x⌉ = ⌈x⌉; otherwise ⌈x⌉ + 1 = ⌈x⌉.
Also
it is clear that the floor of x rounds x down, while the ceiling
of x rounds x up.
iv.Integer Value
Function
The
integer value of x, where x is a real number, converts x
into an integer by truncating or deleting the fractional part of the number and
is denoted by INT (x) or [x].
Example
INT (3.25) = 3
INT (-8.54) = -8
INT (6) = 6
[3.234] = 3
[-6.234] = -6
v.Absolute
Value Function
The
absolute value of x, where x is a real
number is defined as the greater of x or –x and denoted by ABS (x)
= |x|.
If
x is positive, ABS(x) = x; if x is negative, ABS (-x)
= x and ABS (0) = 0.
Example
ABS(-9)
= 9
ABS(3)
= 3
ABS(0)
= 0
vi.Remainder
Function
If
a is any integer and m is
a positive integer, then the integer remainder (function) when a is
divided by m is denoted by a (mod m) [read as ‘a
modulo m’].
Viz.,
a (mod m) is the unique integer r such that a = mq + r,
where 0 ≤r<m.
When
a is positive, we simply divide a by m to get the
remainder r.
Example
30
(mod 5) = 0
3
(mod 5) = 3
30
(mod 7) = 2
When
a is negative, we divide |a| by m and get the remainder r’.
Then
a (mod m) = m – r’ (r’≠0).
Example
-30
(mod 5) = 0
-30
(mod 7) = 7 – 2 = 5
-3
(mod 5) = 5 – 3 = 2
When
two integers a and b have the same remainder when divided by the
positive integer m, then a is said to be congruent to b modulo
m and denoted as
a
≡ b (mod m)
When
a ≡ b (mod m), m divides a – b or a – b is a multiple of m
and hence a =b + km.
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