Is the function f C C Complex Numbers fz z2 injective Surj

Is the function f : C C (Complex Numbers), f(z) = z² injective? Surjective? If you were to define a square root function g, the inverse of f, how would you go about defining it? Can you define g on the entire plane?

Solution

F(z) = z*z where f: C->C

This function is not injective as you will get the value of -9 if you put z=3i or -3i.

This function is surjective as you will get all the complex numbers because square of complex no is a complex number with real part may or may not be zero.

(B) since f is not a bijective function therefore it\'s inverse can\'t be found. Though it\'s inverse can be defined if we make little changes in domain and range. let\'s find out the inverse of f

Y= z²

Therefore z= y^0.5

Now interchange y and z

Y= z^0.5

I.e f^-1=g(z) = z^0.5