Let `f: X->Y`be an invertible function. Show that f has unique inverse. (Hint: suppose `g_1( and g)_2`are two inverses of f. Then for all `y in Y ,fog_1(y)=I_Y(y)=fog_2(y)`Use one oneness of f ).

Let f :X rarr Y be an invertible function . Show that f has unique inverse . (Hint : Suppose g_(1) and g_(2) are two inverse of f. Then for all y inY,(fog_1)(y) =I_(Y) (y) = (fog_(2))(y). Use one - one ness of f).

Let f: X -> Y be an invertible function. Show that the inverse of f^(-1) is f, i.e., (f^(-1))^(-1)= f .

Let f : X rarr Y be an invertible function . Show that the inverse of f^(-1) is f . i.e., (f^(-1))^(-1)= f .

Let f:R->R be a function such that f(x+y)=f(x)+f(y),AA x, y in R.

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all x, then g''f(x) is equal to

Let Y = {n^(2) :n in N} subN Consider f: N rarr Y as f(n) = n^2 . Show that f is invertible. Find the inverse of f.

Let f:R->R and g:R->R be two one-one and onto functions such that they are mirror images of each other about the line y=a . If h(x)=f(x)+g(x) , then h(x) is (A) one-one onto (B) one-one into (D) many-one into (C) many-one onto

Let f be a one-one function with domain {x,y,z} and range {1,2,3} . It is given that exactly one of the following statements is true and the remaining two are false f(X)=1, f(y)!=1 f(z)!=2 determine f^(-1)(1)

Let f and g be differentiable functions satisfying g(a) = b,g' (a) = 2 and fog =I (identity function). then f' (b) is equal to

Let f(x)=x+ sin x . Suppose g denotes the inverse function of f. The value of g'(pi/4+1/sqrt2) has the value equal to