We are given three functions:

• $f(x) = 2x + 5$
• $g(x) = x^2$
• $h(x) = \frac{x}{x - 4}$

We are tasked to find $(f \circ g \circ h)(6)$, which means we first need to compute $h(6)$, then apply $g$ to that result, and finally apply $f$.

Step 1: Calculate $h(6)$

Using the function $h(x) = \frac{x}{x - 4}$, we substitute $x = 6$:

$h(6) = \frac{6}{6 - 4} = \frac{6}{2} = 3$

Step 2: Calculate $g(h(6)) = g(3)$

Next, apply the function $g(x) = x^2$ to the result of $h(6) = 3$:

$g(3) = 3^2 = 9$

Step 3: Calculate $f(g(h(6))) = f(9)$

Finally, apply the function $f(x) = 2x + 5$ to the result of $g(3) = 9$:

$f(9) = 2(9) + 5 = 18 + 5 = 23$

Final Answer:

$(f \circ g \circ h)(6) = 23$

Would you like any further explanation on these steps?

Here are five related questions to expand this topic:

1. How would the result change if we calculated $(g \circ h \circ f)(6)$?
2. What happens if the value of $x$ approaches 4 in the function $h(x)$?
3. Can you calculate $(f \circ h \circ g)(2)$ using the same functions?
4. What are the domains of the functions $f(x)$, $g(x)$, and $h(x)$?
5. How would the function compositions differ if $h(x)$ were defined differently?

Tip: When performing function compositions, work step-by-step from the innermost function outward to avoid mistakes.