If $f(x)=7+4 x$ and $g(x)=\frac{1}{2 x}$, what is the value of $\left(\frac{f}{g}\right)(5)$ ?
The function \(f(x) = 7 + 4x\) is a linear function, and \(g(x) = \frac{1}{2x}\) is a rational function. The expression \(\left(\frac{f}{g}\right)(x)\) is defined as the quotient of the two functions, \(\frac{f(x)}{g(x)}\). To evaluate \(\left(\frac{f}{g}\right)(5)\), we need to compute \(\frac{f(5)}{g(5)}\).
First, we evaluate the numerator \(f(x)\) at \(x = 5\):
\[f(5) = 7 + 4(5) = 7 + 20 = 27.\]
Next, we evaluate the denominator \(g(x)\) at \(x = 5\):
\[g(5) = \frac{1}{2(5)} = \frac{1}{10}.\]
Now, we divide the result of \(f(5)\) by the result of \(g(5)\):
\[\left(\frac{f}{g}\right)(5) = \frac{f(5)}{g(5)} = \frac{27}{\frac{1}{10}}.\]
To divide by a fraction, we multiply by its reciprocal:
\[27 \times 10 = 270.\]
Comparing this result with the given choices (a. \(\frac{11}{2}\), b. \(270\), c. \(160\), d. \(\frac{27}{10}\)), we see that option b is the correct answer.
This problem demonstrates how to combine and evaluate composite functional expressions, which is a fundamental skill in algebra and calculus. In real-world scenarios, such as modeling resource usage, you might have a function representing the total supply \(f(x)\) and another representing the rate of consumption \(g(x)\). Finding the quotient \(\frac{f(x)}{g(x)}\) would then allow you to determine the time remaining until resources are depleted. Understanding how to manipulate these functions ensures you can accurately predict and manage such variables effectively.