Good use of parentheses in asking the question; fellow questioners take note.
[tex]f(x) = \dfrac{x - 7}{x+3}[/tex]
[tex]g(x) = \dfrac{-3x - 7}{x-1}[/tex]
It looks like we have some algebra ahead of us:
[tex]f(g(x)) = f \left( \dfrac{-3x - 7}{x-1} \right) = \dfrac{ \left( \dfrac{-3x - 7}{x-1} \right) - 7}{ \left( \dfrac{-3x - 7}{x-1} \right)+3}[/tex]
When we have a complicated fraction like that it's best if we multiply top and bottom by the inner denominators to simplify:
[tex]f(g(x)) = \dfrac{ \left( \dfrac{-3x - 7}{x-1} \right) - 7}{ \left( \dfrac{-3x - 7}{x-1} \right)+3} \cdot \dfrac{x-1}{x-1} = \dfrac{ (-3x - 7) -7(x-1)}{(-3x - 7)+3(x-1)}[/tex]
[tex]f(g(x)) = \dfrac{ -3x -7 -7x+7}{-3x - 7+3x-3} = \dfrac{-10 x}{-10} = x \quad\checkmark[/tex]
OK, now the other way.
[tex]g(f(x)) = g \left( \dfrac{x - 7}{x+3} \right) = \dfrac{-3 \left( \dfrac{x - 7}{x+3} \right) - 7}{ \left( \dfrac{x - 7}{x+3} \right)-1} \cdot \dfrac{x+3}{x+3}[/tex]
[tex]g(f(x)) = \dfrac{-3(x - 7) - 7(x+3)}{ (x - 7)-(x+3)} = \dfrac{-3x+21-7x-21}{-10} = x \quad\checkmark[/tex]
Not so bad.
