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.
f : N ∴ Y is defined as f(x) = 4 x + 3
where Y = {y ∈ N : y = 4 x + 3 for some x ∈ N }
Consider an arbitrary element y of Y. By the definition of Y, y = 4 x + 3, for some x in the domain N.
Define g : Y 
N given by g (y) = 
Now, g o f(x) = g(f(f))= g(4x + 3) = 
and fo g(y) = f(g(y)) = 
g o f = 1N and f o g = 1y
f is invertible and g is the inverse of f.
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.?
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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