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

If f(x) is an invertible function and g(x)=2f(x)+5, then the value of g^(-1)(x)i s

Consider f: R _+ to [4, oo) given by f (x) = x ^(2) + 4. Show that f is invertible with the inverse f ^(-1) given by f ^(-1) (y) = sqrt (y -4), where R _(+) is the set of all non-negative real numbers.

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 f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)) , then h'(1) is equal to

Let f be twice differentiable function satisfying f(1) = 1, f(2) = 4, f(3) =9 then :

Let f(x) be a polynomial function of second degree. If f (1) =f(-1) and a,b,c are in A.P. then f (a) , f'(b) , f'(c ) are in-

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 _______

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1) =1 , then

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