- The function \(\left(\frac{f}{g}\right)(x)\) is defined as \(\frac{f(x)}{g(x)} = \frac{x^2 - 25}{x - 5}\).
- The domain includes all values of \(x\) where both \(f(x)\) and \(g(x)\) are defined, provided \(g(x)
eq 0\).
- Setting the denominator to zero, \(x - 5 = 0\), reveals the restriction \(x = 5\).
- Therefore, the domain is all real numbers except \(x = 5\), which is option \(\boxed{b}\).
Explanation
- Analyzing the Domain Requirement
The problem asks for the domain of the function \(\left(\frac{f}{g}\right)(x)\), where \(f(x) = x^2 - 25\) and \(g(x) = x - 5\). The definition of the quotient function \(\left(\frac{f}{g}\right)(x)\) is \(\frac{f(x)}{g(x)}\). The domain of this function consists of all real numbers that are in the domain of both \(f(x)\) and \(g(x)\), with the additional requirement that the denominator, \(g(x)\), must not be equal to zero.
- Identifying the Constraint
Both \(f(x) = x^2 - 25\) and \(g(x) = x - 5\) are polynomial functions. Polynomial functions are defined for all real numbers. Thus, the only constraint for the domain of the quotient function \(\frac{f(x)}{g(x)} = \frac{x^2 - 25}{x - 5}\) is that the denominator must be non-zero:
\[g(x)
eq 0\]
\[x - 5
eq 0\]
Solving this inequality for \(x\), we get \(x
eq 5\).
- Determining the Correct Option
Since the functions \(f(x)\) and \(g(x)\) are defined for all real numbers and the only restriction imposed by the quotient is \(x
eq 5\), the domain of the function \(\left(\frac{f}{g}\right)(x)\) consists of all real numbers except \(x = 5\). Comparing this to the given options, this corresponds to option (b).
Examples
In real-life scenarios, calculating the domain is crucial when working with models that describe processes with limitations. For example, if a company models its profit per unit produced by the function \(\frac{P(x)}{n(x)}\), where \(P(x)\) is the total profit and \(n(x)\) is the number of units, the function is undefined if \(n(x) = 0\), representing a state where production is zero. Identifying these "undefined" points helps planners avoid mathematical errors and design systems that only operate within valid, non-zero conditions. In our case, the restriction \(x
eq 5\) ensures that we avoid division by zero, which is a fundamental rule in mathematics.