Show that if `f: A rarr B and g: B rarr C` are one- one, then gof: `A rarrC` is also one-one.

Show that if f: A rarr B and g: B rarr C are onto, then gof: A rarr C is also onto.

If functions f : A rarr B and g : B rarr A satisfy gof = I_A , then show that f is one one and g is onto.

If f : A rarr B and g : B rarr C be the bijective functions , then (gof)^(-1) is ..........

For functions f: A rarr B and g : B rarr A , gof = I_A . Prove that f is one one and g onto functions .

Let A = {-1,0,1,2}, B = {-4,-2,0,2} and f , g : A rarr B be functions defined f(x) = x^(2)-x, x inR and g(x) = 2 |x-(1)/2|-1,x inR . Are f and g equal ? Justify your answer. (Hint : One may note that two functions f : A rarr B and g : A rarr B such that f (a) = g(a) A a inA, are called equal functions).

Show that the VV: RxxR rarr R given by (a, b) rarr max {a, b} and the ^^: RxxR rarr R given by (a, b) rarr min {a, b} are binary operations.

Give examples of two function f: N rarr N and g : N rarr N such that gof is onto but f is not onto. (Hint: Consider f(x) = x+1 " and "g(x) = {x-1 if x>1 1 if x=1.

Give examples of two functions f:N rarr Z and g: Z rarr Z such that gof is injective but g is not injective. (Hint : Consider f(x) = x and g(x) = |x| ) .

Show that the function f: R rarr R , defined as f(x) = x^2 , is neither one-one nor onto.

Find gof and fog, if f:R rarr R and g: R rarrR are given by f(x) = cos x and g(x) = 3x^2 . Show that gof ne fog.