FUNCTIONS

Submitted to:Prof. Shubha Agarwal

Submitted by:Ankit BhandariII year, IV SemIT

WHAT ARE FUNCTIONS?a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Set of input is known as domain, while the set of permissible outputs is called the codomain and the set of all output values of a function.

287

5

36912

15

DOMAIN={2,8,7,5}RANGE={2,6,12}CO-DOMAIN={3,6,9,12,15}

Y=F(x)x y

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45

89

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346

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The function here y=f(x) is not a function because domain of this function has a member that has two image of it in the co domain i.e. 8 has two images, 34 and 5

x yY=f(x)

Function notation is done as f:x->yOr y=F(x)

GRAPHICAL REPRESENTATION OF SETLet F(x)=x+8 be a function, from x->y, then its graph

F(x)=x+8

The graph of y=sin-1 (x) is not a function as the graph of it is intersected by line drawn parallel to y axis more than 1 time

TYPES OF FUNCTION MAPPING1) one-one or injective or monomorphism2) Onto or Surjective3) one-one and onto or bijection4) Many one

One One/injective function

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x yf

Different elements of x have different images in y

Onto function/surjective

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x yf

Many one function

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x yf

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If every element of y is the f image of some element of x is called as onto function

If two or more elements of x have same image in y

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x yf

If the mapping is both one one and onto than it is called as bijective

one-one and onto/bijective

INVERSE FUNCTIONAn inverse function for f, denoted by f−1, is a function in the opposite direction, from Y to XAnd F:y->x

For eg: f(c)=9/5c+32f-1 (F)=5/9(F-32)

As a simple example, if f converts a temperature in degrees Celsius C to degrees Fahrenheit F, the function converting degrees Fahrenheit to degrees Celsius would be a suitable f−1.Such an inverse function exists if and only if f is bijectiveWith this analogy, identity functions are like the multiplicative identity, 1, and inverse functions are like reciprocals

IDENTITY FUNCTIONFunction of the type f(x)=c where c is a constant

They have a line parallel to x axis

POLYNOMIAL FUNCTIONFunctions of the typeF(x)=a0+ a1x+ a2x2…………+ anxn-1

TRIGNOMETRIC FUNCTIONSFunctions like f(x)=sin(x) or f(x)=cos(x)

EXPONENTIAL FUNCTIONFunctions of the type f(x)=ex

The graph of these type of function either remain constant or increase, they never decrease

LOGARITHMIC FUNCTIONSThese functions are inverse of exponential functions and are of form f(x)=loge(x)

COMPOSITE FUNCTIONSfunction composition is the application of one function to the results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x.

Thus one obtains a composite function g ∘ f: X → Z defined by (g ∘ f )(x) = g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f ", or "g round f ", or "g composed with f ", "g after f ", "g following f ", or just "g of f ".

G(f(x))For x=1,G(f(1))=5For x=2,G(f(2))=6