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REAL ANALYSIS
TYPES OF STANDARD FUNCTIONS
STANDARD FUNCTIONS CAN BE CLASSIFIED INTO THREE TYPES:-
1) ALGEBRIC FUNCTIONS
2) TRANSCENDENTAL FUNCTIONS
3) PIECEWISE DEFINED FUNCTIONS
ALGEBRIC FUNCTIONS further classified into:- a) POLYNOMIAL function b) RATIONAL function c) IRRATIONAL function TRANSCENDENTAL FUNCTIONS further classified into:- a) EXPONENTIAL function b) LOGARITHMIC function c) TRIGNOMETRIC function d) INVERSE TRIGNOMETRIC function PIECEWISE DEFINED FUNCTIONS further classified into:- a) CHARACTERISTIC function b) MODULUS function c) SIGNUM function d) INTEGER function 1) ALGEBRIC FUNCTIONS:- a) POLYNOMIAL function :- A function f:D→R where D ⊆R is said to be polynomial
function if f(x)=a 0 + a 1 x +a 2 x^2 +…..+an-1xn-1+anxn^ ∀ x ∈R , where ai ∈R
and an≠0 and n∈N⋃{0}.
DOMAIN=R
RANGE=R (if n is odd) =A proper subset of R (If n is even) NOTE:- if n is even and an>0 than range is [M,∞) and if an<0 than range is (-∞,M]. Here, R is real number space. b) RATIONAL function :- A function f:D→R where D ⊆R is said to be rational function if f(x)=p(x)/q(x) where p(x) and q(x) are polynomial function.
DOMAIN=R / { x ∈R /q(x)=0}
c) IRRATIONAL function :- A function f:D→R where D ⊆R is said to be irrational
function s.t. f(x)=x1/n^ , where n∈N, n>1.
DOMAIN= R , if n is odd. = [0,∞) , if n is even. RANGE= R , if n is odd. = [0,∞) , if n is even. 2) TRANSCENDENTAL FUNCTIONS :- a) EXPONENTIAL function :- A function f:D→R where D ⊆R is said to be exponential function if f(x)=ax^ where a>0,a≠1. Where ‘a’ is base and ‘x’ is called exponent. DOMAIN= R RANGE =(0,∞) b) LOGARITHMIC function:- A function f:D→R where D ⊆R is said to be logarithmic function if f(x)=loga(x) where a>0. DOMAIN= (0,∞) RANGE = R. c) TRIGNOMETRIC function :- A function f:D→R where D ⊆R is said to be trignometric function s.t TRIGNOMETRIC function DOMAIN RANGE Sin(x) R [-1,1] Cos(x) R [-1,1]
Tan(x) R[(2n+1)π/2 ;n∈
Z]
R
Cot(x) R[nπ ;n∈ Z] R
Sec(x) R[(2n+1)π/2 ;n∈
Z]
(-∞,1] ⋃
[1,∞)
Cosec(x) R[nπ;n∈ Z] (-∞,1] ⋃
[1,∞) where ‘Z’ is space of all integers. d) INVERSE TRIGNOMETRIC function:- A function f:D→R where D ⊆R is said to be inverse trignometric function s.t INVERSE TRIGNOMETRIC function DOMAIN RANGE Sin-1(x) [-1,1] [-π/2, π/2] Cos-1(x) [-1,1] [0, π] Tan-1(x) R (-π/2, π/2) Cot-1(x) R (0, π)
Sec-1(x) (-∞,1] ⋃ [0, π] { π/2}
Or
[x]=n; n≤x≤n+1 ∀ n∈ Z
DOMAIN= R
RANGE = Z
For example:-[2.3] = [-3.8] =- Where ‘Z’ is space of all integers. LEAST INTEGER function:- A function f:D→R where D ⊆R is said to be least integer function s.t. (x)=the least integer that is less than equal to x. Or
(x)=n; n-1≤x≤n ∀ n∈Z
DOMAIN= R
RANGE = Z
For example:-(2.3) = (-3.8) =- Where ‘Z’ is space of all integers.
