Types of Functions.Functions and different types of functions. Constant, Identity, Equality, Polynomial, Power, Rational, Irrational, Absolute (Modulus), Signum, Exponential, Integer, Monotonic, Periodic. Composition, Strictly Monotonic, Increasing, Decreasing

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__Constant Function__:

A function f :X->Y is called a

**constant function**if its range is a singleton set i.e., f(x) = c for all x E X where c is some constant.NOTE : For a function of real variable

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__Identity Function__:

A function f:X->X is called an

**Identity Function**if##
__Equality of Function__:

Two functions f:X->Y and g:X->Y are said to be equal iff f(x)=g(x) for all x E X i.e., iff D

_{f}= D_{g}and R_{f}= R_{g.}##
_{Polynomial Function:}

_{The function f(x) = a0 + a1x +...+anxn , where a0,...,an are real constants such that an not equal to 0 and "n" is a positive integer, is called a Polynomial Function in "x" with real co-efficients and it is of degree "n".}

_{ }

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_{Power Function:}

_{A function f: R->R defined by, f(x)=xa, a E R is called a power function.}

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_{Rational Function:}

_{A function defined by the quotient of two polynomial functions is called a rational function. Thus R(x) = P(x)/Q(x), where P(x) and Q(x)[q!=0]are polynomial functions, is called a rational function.}

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_{Irrational Function:}

_{A function involving one or more radicals of polynomials is called an irrational function}

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_{Absolute value Function:}

_{The absolute value function or the numerical value function or the modulus (mod) of a real number x, denoted by |x|, is defined as }

_{|x| = x if x>0}

_{or, |x| = -x if x<0 font="">}

_{or, |x| = 0 if x = 0}

_{Properties of Modulus of a real number}

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_{ The function f : R->R defined by f(x) = |x|, is called Modulus Function.}

_{Here, Df = R, Rf = R+}

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_{Signum Function:}

_{The Signum of the real number x is defined as}

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_{
}

_{The function f:R->R given by f(x) = Sgn(x) is called a Signum function. Here Df and R(f)= {-1, 0 , 1}.}

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_{Greatest Integer Function:}

_{For all x E R. Let [x] denote the greatest integer in x not greater integer in x not greater than x.}

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_{The function f: R->R given by f(x) = [x] is called greatest integer function or the step function.}

_{Note. Here Df = R and Rf = Z, set of all integers.}

_{ }

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_{Exponential Function:}

_{The function f:R->R defined by f(x) = ax, where "a" is a positive real number different from 1, is called an exponential function.}

_{Here Df = R and Rf = R+ - {0}.}

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_{Logarithmic Function:}

_{Let f : R+ - 0 ->R be defined by f(x) = logax where a[not equal to 0], and a[not equal to]. Then f is called a logarithmic function.}

_{Here Df = R+ - {0}, Rf = R.}

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_{Periodic Function:}

_{A function y=f(x) defined on a domain X is periodic if there exists a constant a>0 such that f(x+a) = f(x-a) =f(x) for all x E X.}

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_{Monotonic Function:}

_{Let f: [a,b]->R be a function. The function f is said to be}

iv. ~~V~~ x

**strictly monotonic decreasing**on [a,b] if_{1},x_{2}E[a,b],##
__Composition of functions:__

Let f : X-> Y be defined by y=f(x)

and g: X-> Z be defined by z= g(y)

then, h: X->Z defined by h(x) = g(f(x)) is called

**composite function**.
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