In order to calculate f - g, we need to subtract the functions. To do so, we will need to put the fractions in the same denominator.
So we have:
[tex]\begin{gathered} f-g \\ =\frac{x-1}{x+3}-\frac{x-3}{x+1} \\ =\frac{(x-1)(x+1)}{(x+3)(x+1)}-\frac{(x-3)(x+3)}{(x+1)(x+3)} \\ =\frac{x^2-1}{(x+3)(x+1)}-\frac{x^2-9}{(x+3)(x+1)}_{} \\ =\frac{x^2-1-(x^2-9)}{(x+3)(x+1)} \\ =\frac{x^2-1-x^2+9}{(x+3)(x+1)} \\ =\frac{8}{(x+3)(x+1)} \end{gathered}[/tex]
