X and Y are two sets and `f: X->Y` If `{f(c)=y; c subX, y subY}` and `{f^(-1)(d)=x;d subY,x sub X`, then the true statement is `(a) f(f^(-1)(b))=b` `(b) f^(-1)(f(a))=a` `(c) f(f^(-1)(b))=b, b sub y` `(d) f^(-1)(f(a))=a, a sub x`

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

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)-|f(x)| (d) none of these

If f(x)=log((1+x)/(1-x)),t h e n (a) f(x_1)f(x)=f(x_1+x_2) (b) f(x+2)-2f(x+1)+f(x)=0 (c) f(x)+f(x+1)=f(x^2+x) (d) f(x_1)+f(x_2)=f((x_1+x_2)/(1+x_1x_2))

The domain of f(x)i s(0,1)dot Then the domain of (f(e^x)+f(1n|x|) is (a) (-1, e) (b) (1, e) (c) (-e ,-1) (d) (-e ,1)

A function f: RvecR satisfies the equation f(x)f(y)-f(x y)=x+yAAx ,y in Ra n df(1)>0 , then f(x)f^(-1)(x)=x^2-4 b. f(x)f^(-1)(x)=x^2-6 c. f(x)f^(-1)(x)=x^2-1 d. none of these

Let f be a continuous, differentiable, and bijective function. If the tangent to y=f(x)a tx=a is also the normal to y=f(x)a tx=b , then there exists at least one c in (a , b) such that (a) f^(prime)(c)=0 (b) f^(prime)(c)>0 (c) f^(prime)(c) (d) none of these

If f(x+1/2)+f(x-1/2)=f(x) for all x in R , then the period of f(x) is 1 (b) 2 (c) 3 (d) 4

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in Rdot If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to (a)6 (b) 0 (c) 3f(3) (d) 2f(0)

Which of the following functions is the inverse of itself? (a) f(x)=(1-x)/(1+x) (b) f(x)=5^(logx) (c) f(x)=2^(x(x-1)) (d) None of these

If f(x) is monotonic differentiable function on [a , b] , then int_a^bf(x)dx+int_(f(a))^(f(b))f^(-1)(x)dx= (a) bf(a)-af(b) (b) bf(b)-af(a) (c) f(a)+f(b) (d) cannot be found