Type of functions:
One - One or injective function: A function f: X → Y is called to one - one or injective if every element in the domain of a function has a different image in the co - domain. Example: f:R→R f(x) = 2x is one - one.
Many - one function: A function f: X →Y is called be many one if there are at least two components in the domain whose images are the exact.
Example: f: R→R shown by f(x) = x2 is Many - one.
METHODS TO CALCULATE ONE - ONE AND MANY - ONE:
- If f(x1) = f (x2) => x1 = x2 for each and every x1,x2 in the domain, then 'f' is one - one else many - one.
- If the function is completely decreasing or increasing in the domain, then 'f' is one - one else many - one.
Graphical Method: If we define a line parallel to the x - axis cut the graph of y = f(x) at one and only single point, then f(x) is one - one and if the line parallel to the x - axis intersect the graph at more than one distinct points then f(x) is a many - one function.
- Any continuous function f(x) which have at least one local minima or local maxima is many - one.
- All even functions have to be many one.
- All polynomials of even degree distinct on R have at least one local maximum or minima and therefore are many one on the domain R. Polynomials of odd degree may be one - one or many - one.
Onto function or Surjective function: A function f: X→Y is called be a onto function or Surjective function if and only if every element of Y is the image of any element of X i. e. if and only if for each y∈Y there exists some x∈X such that y = f(x). Therefore 'f' is onto if f(x) = Y i. e. range = co - domain of function.
Example: The map f: R →[ -1,1] provided by f(x) = sinx is an onto map.
Into function: A function f: X→Y is called an into function if there exists at least one component in the co - domain Y which is not an reflection of any element in the domain X.
Example: The map f: R →R provided by f(x) = x2 is an into map
One - one onto map or bijective function: A function f: X →Y is called one - one onto or bijective function if and only if
(i) Different elements of X have different images in Y
(ii) Each element of Y has at least one pre - image in X.
Example: The map f: X→Y provided by f(x) = 2x is a one - one onto map.
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