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Saturday, February 13, 2016

Types of functions


Even Function:
If f(x) = f(-x) then we call that function as Even fucntion.  Even functions are symmetric around Y axis
Eg: f(x)=x21 is an even function.

Odd Function:

If f(-x) = -f(x) then we call that function as Odd function.  In the graph of odd function, the first and third quadrants will be reflections of each other and so will the second and fourth quadrant.
Eg: f(x)=x3x

Modulus function: 
The function defined by f(x) = |x|  ={xxx0x<0  is called the modulus function. The Domain of |x| = R and Range = R+ U {0}, i.e. set of all non-negative real numbers. 

Greatest integer function: 
For any real number x, we denote [x], the greatest  integer less than or equal to x. For example,
[2.45] = 2 and [– 2.1] = – 3. The function defined by f(x) = [x] for all x  R is called the greatest integer function. Obviously, the Domain of [x] = R and the Range = Z, i.e. set of integers.

Square root function:
A function f(x) defined by f(x) = x,  xR+ is called the square root function. The Domain and Range of the square root function is [0, ) which is set of all non-negative real numbers.

Signum function:
The function defined by  f (x) = {|x|x0x0x=0{110x>0x<0x=0 is called the signum function also given by sgn(x). The Domain = R and Range = {– 1, 0, 1}.

Polynomial function:
A function of the form f(x) = a0xn+a1xn1+......+an1xn1+an,  where  a0,a1...an an are real numbers,  a00and nN, is called a polynomial function of degree n.   The Domain of a polynomial function is R. Range can be any subset of real numbers.

Reciprocal function:
The function that associates each non-zero real number x to its reciprocal, i.e. f(x) = 1x  is called the reciprocal function. Domain = Range = R – {0}.

Exponential function:
The function that associates every real number x to ex , i.e. f(x) = ex for all x  R, is called the exponential function.Domain = R and Range = R+ , i.e. set of positive real numbers.

Logarithmic function: 
The function that associates every positive real number x to log x, i.e. f(x) = log x for all x > 0 is called the logarithmic function . Domain = R+ and Range = R. 

Composite functions:
Consider two functions f(x) = x21 and g(x) = 5 - 2x
The expressions of the type  f(g(x)) are called composite function.
f(g(x)) = f(5-2x) = (52x)21
f(g(x)) need not be equal to g(f(x))

Iterative Function:
In the previous example we performed function g on f. Similarly performing f on f is called iteration.
Thus f(f(x) = f(x21) = (x21)21
Extending this idea further, one could also have computed fn(x) = f(f(f(.........f(x)....)
This is called a nested function or iterative function.
Thus f4(x)=f(f(f(f(x))))
If f(x) = 2x - 1
f(f(x)) = 2 (2x-1) - 1 = 4x - 3

Periodic function:
If a function f(x), repeats its value after a definite increment (or decrement) in the value of x, then we say that the function f(x) is a periodic function. 
F is periodic then
f(x+p) = f(x) for p is a real number.

Example: Given that f(x-a) + f(x) = 0.  Find that least possible number P such that f(x+P) = f(x)
Sol: f(x-a) = -f(x)
replacing x by (x-a)
f(x-2a) = -f(x-a) = f(x) [as f(x-a) = -f(x)]
replacing x by x+2a
f(x) = f(x+2a)
So the minimum possible value is 2a and it is the period of the function f(x)

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