PLZ HELP! URGENT! functions/inverse functions?

Question

Ive been stuck on this for a day could someone PLEASE help?? PLEASE SHOW WORKING SO I CAN UNDERSTAND!!!

The function 'g' is defined by :

g : x ----> 4/(1+x^2)

a) Give a suitable domain for 'x' so that 'g' is a one-to-one function and state the range of 'g' for this domain.

b) Define an inverse function g^-1 stating its domain and the corresponding range.

Answer 1

a) Take 0 <= x < inf. Then g is one-to-one. The range is 0 < g(x) < = 4.

b) Solve g(f(x)) = x to find f(x) = sqrt(4/x - 1); this is g^(-1). The domain is 0 < x <=4, and the range is 0 <= g^-1)(x) < inf.