Find the domain of $f(x)=\frac{7 x}{x-3}$.
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a rational function, which is a ratio of two polynomials, the function is undefined when the denominator is zero. Given the function \(f(x) = \frac{7x}{x-3}\), we must ensure that the denominator, \(x-3\), is not equal to zero to avoid division by zero.
To find the values for which the function is undefined, we set the denominator equal to zero:
\[x - 3 = 0\]
Adding 3 to both sides of the equation yields:
\[x = 3\]
This means the function \(f(x)\) is undefined when \(x = 3\). Therefore, the domain of \(f(x)\) includes all real numbers except for \(x = 3\).
We have determined that the function \(f(x) = \frac{7x}{x-3}\) is defined for all real numbers except \(x = 3\). Comparing this result with the provided options, the correct answer is the one that states "All real numbers except \(3\)".
Knowing the domain of a function is crucial in real-world applications, such as calculating the cost-per-item in a manufacturing process. If the cost function is defined as \(C(n) = \frac{Total Cost}{n}\), where \(n\) is the number of items, we know that the domain cannot include \(n=0\) because we cannot divide by zero items. Similarly, in electrical engineering, the impedance of certain circuits might involve rational functions where certain frequencies could lead to undefined resistance, which would be identified by determining the function's domain.