Short Answer TypeLet f : x → Y be an invertible function. Show that the inverse of f–1 is f,
i.e.(f–1)–1 = f.
Let f : N → Y be function defined as f (x) = 4 x + 3, where, Y = {y ∈N : y = 4 x + 3 for some x ∈ N}. Show that f is invertible. Find the inverse.
Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer. (A) (2, 4) ∈ R (B) (3, 8) ∈ R (C) (6, 8) ∈ R (D) (8, 7) ∈ R
Show that the function f : R. → R. defined by 
is one-one and onto, where R. is the set of all non-zero real numbers. Is the result true, if the domain R. is replaced by N with co-domain being same as R.?
It is given that R. → R. by 
one - one
f(x) = f(y)
x = y
f is one- one onto
It is clear that 
, there exists 
(Exiasts as y 
0) such that f(x) = 
= y
f is onto
Thus the given function (f) one-one and onto
Now, consider function g : N 
R defined by g(x)= 
We have g (x1) = g(x2) 
g is one - one
Further, it is clear that g is not onto as for 1.2 
R. there does not exit any x in N such that g(x) = 
hence function g is one-one but not onto.
Check the injectivity and surjectivity of the following functions:
f : N → N given by f(x) = x2
Check the injectivity and surjectivity of the following functions:
f : Z → Z given by f(x) = x2
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