Assume that $$f:(-2, -1) \rightarrow (1, 2)$$ is an onto function and for i = 1, 2, 3, 4 , define $$g_{1}(x) = f(x) -2, g_{2}(x) = f(-x), g_{3}(x) = -f(-x)$$ and $$g_{4}(x) = f(-x-2)$$. What is the correct arrangement of $$g_{1}, g_{2}, g_{3}, g_{4}$$ such that the graph of the $$k^{th}$$ function lies in the $$k^{th}$$ quadrant for k= 1, 2, 3, 4?

Here we need to determine to which quadrant each of the g1, g2, g3, g4 goes to. For that we need to find the range and domain of these functions. The domain of g1 is same as that of f as the value is directly inserted in to the function. Since the value of g1(x) is 2 less than the f(x) the range would be(-1,0). So g1 goes to 3rd quadrant. Similarly for g2 the domain should be negative of what f has as the input for f is