Let `f:X to Y` be an invertible function. Show that f has uniques inverse.

Let f : X to 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 differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______

If f: R->R is an invertible function such that f(x) and f^-1(x) are symmetric about the line y=-x, then (a) f(x) is odd (b) f(x) and f^-1(x) may be symmetric (c) f(c) may not be odd (d) none of these

Let g(x) be the inverse of an invertible function f(x) which is differentiable at x=c . Then g^(prime)(f(x)) equal. (a) f^(prime)(c) (b) 1/(f^(prime)(c)) (c) f(c) (d) none of these

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^('')(f(x)) equals. (a) -(f^('')(x))/((f^'(x))^3) (b) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (c) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

Let f be continuous and differentiable function such that f(x) and f'(x) has opposite signs everywhere. Then (A) f is increasing (B) f is decreasing (C) |f| is non-monotonic (D) |f| is decreasing

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

Let f: N->Z be a function defined as f(x)=x-1000. Show that f is an into function.

Let f : R to R be a function such that f(0)=1 and for any x,y in R, f(xy+1)=f(x)f(y)-f(y)-x+2. Then f is

Let f: RrarrR, where f(x)=sinx , Show that f is into.