We are given two functions, $f(x) = \frac{x - 4}{4 - x}$ and $g(x) = 2x - 5$, and the task is to find the inverse of $(g \circ f)(x)$, which is the composition of the functions $g(f(x))$.

Step 1: Find $g(f(x))$

The composition $g(f(x))$ means applying $g$ to the result of $f(x)$. So, substitute $f(x)$ into $g(x)$: $g(f(x)) = g\left(\frac{x - 4}{4 - x}\right)$ Since $g(x) = 2x - 5$, replace $x$ with $\frac{x - 4}{4 - x}$: $g(f(x)) = 2\left(\frac{x - 4}{4 - x}\right) - 5$ Now simplify this expression: $g(f(x)) = \frac{2(x - 4)}{4 - x} - 5$ The first term becomes: $\frac{2(x - 4)}{4 - x} = \frac{2x - 8}{4 - x}$ Thus: $g(f(x)) = \frac{2x - 8}{4 - x} - 5$ Convert $-5$ to a fraction with the same denominator: $g(f(x)) = \frac{2x - 8}{4 - x} - \frac{5(4 - x)}{4 - x} = \frac{2x - 8 - 20 + 5x}{4 - x}$ Simplify the numerator: $g(f(x)) = \frac{7x - 28}{4 - x}$

Step 2: Find the Inverse of $g(f(x))$

To find the inverse of $g(f(x))$, set $y = \frac{7x - 28}{4 - x}$ and solve for $x$ in terms of $y$: $y(4 - x) = 7x - 28$ Expand both sides: $4y - yx = 7x - 28$ Collect all terms involving $x$ on one side: $4y + 28 = 7x + yx$ Factor $x$ on the right-hand side: $4y + 28 = x(7 + y)$ Solve for $x$: $x = \frac{4y + 28}{7 + y}$

Step 3: Verify the Answer

This expression matches option A: $\frac{7x + 28}{x + 4}$

Thus, the correct answer is A.

Would you like more details or explanations about any step? Here are some related questions you could explore:

1. What is the domain of the function $f(x)$?
2. How do you verify if a function is invertible?
3. What is the graphical interpretation of the inverse function?
4. How can we simplify complex fractions in function compositions?
5. What are the properties of composite functions?

Tip: When solving for the inverse of a function, always remember to express the original variable $x$ in terms of the new variable $y$, and then swap them to finalize the inverse.