Bijections
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A function f: x èy is said to be a Surjective function or an Onto function if for every y E Y, there exists at least one x E X such that y = f(x). In an onto function, the range and co-domain become equal.
A function f : x èy is said to be an Injective function or a one-to-one function if for all x1, x2E X,
f(x1) = f(x2)
==> x1 = x2
In other words, if f is one-to-one, each value in the range can have only one preimage in the domain.
A function that is both surjective and Injective is called a Bijective function or a Bijection.
From the definitions of one-to-one and onto functions, we can see that if f: x èy is a bijection, then for each y E Y, there should be exactly one x E X such that y = f(x). In other words, each y in the co-domain should have exactly one preimage in the domain.
From a graph if we want to find the value of x for which y = f(x) = y1, then we draw a horizontal line through y1 (i.e. the line y = y1) till it intersects the graph and then read the value of x by dropping a perpendicular from that point to the x - axis.
So if we want to check whether or not a function f: R ==>R is a bijection, using the graph of y = f(x), then
(a) If f is a bijection, then every horizontal line will cut the graph of the function at exactly one point ( because for each y E R we must have exactly one x E R such that y=f(x) )
(b) If for y = y1, the horizontal line y = y1, cuts the graph at two or more points, then the function is not one - to - one ( because there are two values of x, say x1 and x2, such that f(x) = y1.
i.e. f(x1) = f(x2) = y1 but x1 <> x2)
(c) If for some y = y2, the horizontal line y = y2does not cut the graph at all, then the function is not onto (because then there is not x such that f(x) = y2and y2E Y, the co-domain)
THE INVERSE OF A FUNCTION
The functions f(x) = x1/3 and g(x) = x3 have the following property
-
f(g(x)) = f(x3) = (x3)1/3 = x
-
g(f(x)) = g(x1/3) = (x1/3)3 = x
Similarly the functions Sin x and sin-1x cancel the effect of one another. Thus, if two functions f and g satisfy f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f, we say that f is the inverse of g and g is the inverse of f. We can write this as f = g-1 and g = f-1
COMPOSITE FUNCTIONS
Let f: X èY and g: Y ==>Z be two functions. Then we can define a function h: X ==>Z such that h(x) = g (f (x) ) .
Then h is called the composite function of g and f and we write h = g o f.