Argh ok a couple of clues here would be useful !

Let f(z) = 5i(z + i)

1 - Describe the effect f has on a typical point of C in terms of geometric transformations.

Is it that it is scaled by a factor of 5, moved up 1 unit and rotated by 90 degrees ?

2 - Let GAMMA be the path with the parametrization

gamma(t) = 2 - i + 3e^it

Sketch GAMMA indicating the direction of increasing t and identifying
its initial and final points in cartesian form.

So I change 3e^it to 3(cost + isint) ?

So I get 2-i+3(cost + isint)

Then do i need to get the real and imaginary parts for the parametric equations I think ?

3 - By applying your geometric description of the function f to GAMMA, sketch the path f(GAMMA). Indicate the direction of f(GAMMA) and identify it's inital and final points in Cartesian form.

4 - Write down the standard parametrization of f(GAMMA)

Bit stuck on 3 and 4.

5 - Let f(z) = exp(1-2iz)
g(z) = Log(1-2iz)
h(z) = (1-2iz)^3i

a - write down the domain of each of the functions f,g and h

f would be all complex numbers.

In g the part 1-2iz would have to be positive but I'm not sure how I describe that.

Similar case for h

b - Explain why f does not have an inverse fucntion.

Because of the periodicity e^z+2piei = e^z

c - Show that g has an inverse function and find it's domain and rule

erm ?!?

Any clues pointers appreciated cheers !