1. The user has provided multiple function definitions and asked for compositions $g \circ f(x)$, $g \circ h(x)$, and $f \circ h(x)$ where: $$ f(x) = \frac{x}{x-1}, \quad g(x) = 3x - 5, \quad h(x) = x^2 $$ 2. Calculate $g \circ f(x)$: $$ g(f(x)) = g\left(\frac{x}{x-1}\right) = 3\cdot\frac{x}{x-1} - 5 = \frac{3x}{x-1} - 5 = \frac{3x - 5(x-1)}{x-1} = \frac{3x - 5x + 5}{x-1} = \frac{-2x + 5}{x-1} $$ 3. Calculate $g \circ h(x)$: $$ g(h(x)) = g(x^2) = 3(x^2) - 5 = 3x^2 - 5 $$ 4. Calculate $f \circ h(x)$: $$ f(h(x)) = f(x^2) = \frac{x^2}{x^2 - 1} $$ 5. Summary: - $g \circ f(x) = \frac{-2x + 5}{x - 1}$ - $g \circ h(x) = 3x^2 - 5$ - $f \circ h(x) = \frac{x^2}{x^2 - 1}$ These compositions follow the rules of substitution where the output of inner function becomes the input to the outer function.