- Identify the denominator of the rational function as \(4x + 4\).
- Set the denominator equal to zero to find the points where the function is undefined: \(4x + 4 = 0\).
- Solve the linear equation to get \(x = -1\).
- Conclude that the domain is all real numbers except \(-1\), so the excluded value is \(\boxed{-1}\).
Explanation
- Analyzing the Function
To find the domain of the function \(f(x) = \frac{4x^2 + 16x + 4}{4x + 4}\), we must determine the set of all real numbers \(x\) for which the expression is well-defined. A rational function, which is a fraction of two polynomials, is defined for all real numbers except those that make the denominator equal to zero. In this case, the numerator is \(4x^2 + 16x + 4\) and the denominator is \(4x + 4\).
- Identifying the Excluded Value
The condition for the function to be undefined is when the denominator is equal to zero. We set the denominator expression to zero to identify the excluded values:
\[4x + 4 = 0.\]
Subtracting 4 from both sides of the equation, we get:
\[4x = -4.\]
Dividing both sides by 4, we find the excluded value:
\[x = -1.\]
- Defining the Domain
Since the denominator is zero when \(x = -1\), the function \(f(x)\) is not defined at this point. Therefore, the domain of the function is all real numbers except \(-1\). Expressed in set-builder notation, the domain is \(\{x \in \mathbb{R} \mid x
eq -1\}\). As requested, the value not in the domain, expressed as a reduced fraction, is \(-1/1\), which simplifies to \(-1\). Thus, the only value of \(x\) not in the domain is \(-1\).
Examples
In real-world applications, identifying the domain of a function is crucial for modeling scenarios like production costs or population growth. For example, if a cost function for a factory is defined by \(C(n) = \frac{1000}{n - 50}\), where \(n\) is the number of units produced, the function is undefined when \(n = 50\). This tells the manager that the formula cannot be used at that specific production level because it would lead to division by zero, representing a breakdown or a singularity in the model. Understanding where a function is undefined helps avoid such errors in calculations.