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

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.

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 f: R rarr R , f(x) = x/(x^(2)+1) is not one one and onto function.

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.

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

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| ) .

Let the function f : R rarr R be defined by f(x) = cos x , AA x in R . Show that f is nether one - one nor onto .

Show that the function f : R rarr {x inR:-1lt x lt1} defined by f(x) =x/(1+|x|),x in R is one one and onto function.

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 f: R_(+) rarrR_(+), f(x) =1/x is one to one and onto function.