Need help with surjective/injective

Question

Consider the function f:C→C

defined by f(x+iy)=(3x+2)+i(y²−1) where x,y are elements of R

.

1. Is the function injective? Explain your reasoning.
2. Is the function surjective? Explain your reasoning.
3. Find range of f

.

Answer 1

The function is neither injective nor surjective.

1. f takes the same value at x+iy and x–iy, so it is not injective.
2. the imaginary part of f is y²–1, which means f only takes value in the half-plane above the line z=–1.
3. To determine the range of f, let f(x+iy)=u+iv, where u, v are real. Then 3x+2=u, and y²–1=v. We have x=(u-2)/3, and y=±√(v+1). We can see easily that u can take any real value but v has to be ≥ -1. So the range of f is the half-plane above the line v=-1(inclusive).