- The composition of functions \((r \circ q \circ z)(x)\) is defined as \(r(q(z(x)))\).
- We first evaluate the innermost function, \(z(x) = \frac{8}{x-8}\).
- Next, we evaluate \(q(z(x)) = e^{\left(\frac{8}{x-8}\right)}\).
- Finally, applying the outermost function \(r(x) = \sqrt{x}\) yields \(\sqrt{e^{\left(\frac{8}{x-8}\right)}}\), which corresponds to the first option.
- The final answer is \(\boxed{\sqrt{e^{\left(\frac{8}{x-8}\right)}}}\).
Explanation
- Analyzing the Given Functions and Composition
To determine the correct expression for the composition of functions \((r \circ q \circ z)(x)\), we must first analyze the definitions of the functions provided:
- \(q(x) = e^x\)
- \(r(x) = \sqrt{x}\)
- \(z(x) = \frac{8}{x-8}\)
The notation \((r \circ q \circ z)(x)\) indicates a nested composition of these three functions, which is evaluated from the innermost function outward:
\[(r \circ q \circ z)(x) = r(q(z(x))).\]
This means we apply \(z(x)\) first, then apply \(q(x)\) to the output of \(z(x)\), and finally apply \(r(x)\) to that result.
- Calculating the Composite Function Expression
We start with the innermost function, \(z(x) = \frac{8}{x-8}\).
Next, we substitute this into the function \(q(x) = e^x\). This gives us:
\[q(z(x)) = q\left(\frac{8}{x-8}\right) = e^{\left(\frac{8}{x-8}\right)}.\]
Now, we take this intermediate result and substitute it into the outermost function \(r(x) = \sqrt{x}\):
\[r(q(z(x))) = r\left(e^{\left(\frac{8}{x-8}\right)}\right) = \sqrt{e^{\left(\frac{8}{x-8}\right)}}.\]
- Conclusion and Final Result
We have found that the composition \((r \circ q \circ z)(x)\) is equal to \(\sqrt{e^{\left(\frac{8}{x-8}\right)}}\). Comparing this to the given options:
- \(\sqrt{e^{\left(\frac{8}{x-8}\right)}}\)
- \(\frac{8}{\sqrt{e^x}-8}\)
- \(e^{\left(\frac{8}{\sqrt{x}-8}\right)}\)
- \(\frac{8}{e^{\sqrt{x}}-8}\)
The derived expression matches the first option. Thus, the correct expression for the composite function \((r \circ q \circ z)(x)\) is \(\boxed{\sqrt{e^{\left(\frac{8}{x-8}\right)}}}\).
Examples
Composite functions are frequently used in modeling complex systems where one process depends on the outcome of another. For example, consider a supply chain system where the cost of raw materials depends on the current market price \(z(x)\), the production process then modifies that cost based on an exponential growth model \(q(x)\), and the final retail price is adjusted by a square root scale factor \(r(x)\) due to bulk-ordering efficiencies. In this scenario, evaluating the composition of these functions \((r \circ q \circ z)(x)\) allows an analyst to determine the final retail price directly from the initial market price of the raw materials, streamlining the calculation process.