These two functions,  f(x)  and  g(x),  have different domain;  

in other words, they are defined in different sets of real numbers.


The domain for function g(x)  is  { x |  x > 1 }.


The domain for function f(x)  is  the set of solutions to THIS inequality

     > 0,   


which is the UNION of two sets

    { x < -1 } U { x > 1 },


where the numerator and denominator BOTH are negative OR BOTH are positive.



See this visual illustration, where I slightly changed function g(x) intently to make the difference visible.


    


    Plot  f(x) =  (red), g(x) =  (green)


    f(x)  has two branches, as you see (two red lines);  

    g(x)  has only one branch (only one green line).