In the problem, you are given the following information:

1. $f(x) = 2x - 1$
2. $g(x) = \frac{5x}{x + 1}$
3. $(g \circ h)(x) = x - 2$

You are asked to find $(h \circ f)(x)$.

Step-by-Step Solution:

1. Understanding the composite function $(g \circ h)(x)$: $(g \circ h)(x) = g(h(x)) = x - 2$ This means that $g(h(x)) = \frac{5h(x)}{h(x) + 1} = x - 2$.

2. Solve for $h(x)$: Set up the equation using the expression for $g(h(x))$: $\frac{5h(x)}{h(x) + 1} = x - 2$ Now, cross-multiply to solve for $h(x)$: $5h(x) = (x - 2)(h(x) + 1)$ Expand the right-hand side: $5h(x) = (x - 2)h(x) + (x - 2)$ Bring all terms involving $h(x)$ to one side: $5h(x) - (x - 2)h(x) = x - 2$ Factor out $h(x)$: $h(x)(5 - (x - 2)) = x - 2$ Simplify the expression: $h(x)(7 - x) = x - 2$ Solve for $h(x)$: $h(x) = \frac{x - 2}{7 - x}$

3. Find $(h \circ f)(x)$: Now, substitute $f(x) = 2x - 1$ into $h(x)$: $(h \circ f)(x) = h(f(x)) = h(2x - 1)$ Substitute $2x - 1$ into the expression for $h(x)$: $h(2x - 1) = \frac{(2x - 1) - 2}{7 - (2x - 1)}$ Simplify the numerator and denominator: $h(2x - 1) = \frac{2x - 3}{8 - 2x}$

Thus, the expression for $(h \circ f)(x)$ is: $\frac{2x - 3}{-2x + 8}$

Answer:

The correct choice is D.

Would you like more details or have any further questions? Here are five related questions to expand on this:

1. What is the definition of a composite function, and how does it work?
2. How would the solution change if $g(x)$ were different?
3. Can you explain the process of solving for $h(x)$ in more detail?
4. How can we verify that our solution for $h(x)$ is correct?
5. What are some common mistakes when working with composite functions?

Tip: When solving for a function within a composite, always carefully handle the algebraic manipulation to avoid errors.