Which of the following describes the domain of the piecewise function [tex]g(x)=\left\{\begin{array}{lll}\frac{x^2+4 x}{x^2+2 x-8} & \text { for } & x\ \textless \ 4 \\ \log _3(x+5) & \text { for } & x \geq 4\end{array} ?\right.[/tex]

A. [tex](-\infty, 2) \cup(2,4) \cup(4, \infty)[/tex]
B. [tex](-\infty,-4) \cup(-4,2) \cup(2, \infty)[/tex]
C. [tex](-\infty, 2) \cup(2, \infty)[/tex]
D. [tex](-\infty, \infty)[/tex]

Determine the domain of the rational function part: ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , 4 ) .
Determine the domain of the logarithmic function part: [ 4 , ∞ ) .
Combine the domains: ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , 4 ) ∪ [ 4 , ∞ ) .
Simplify the union of intervals: ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , ∞ ) .

The domain of the piecewise function is ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , ∞ ) ​ .
Explanation

Understanding the Problem We are given a piecewise function and asked to determine its domain. The domain is the set of all possible x values for which the function is defined. We need to consider the domain of each piece separately and then combine them.

Analyzing the Rational Function The first piece is a rational function: g ( x ) = x 2 + 2 x − 8 x 2 + 4 x ​ for x < 4 . Rational functions are defined everywhere except where the denominator is zero. So, we need to find the values of x for which x 2 + 2 x − 8 = 0 .

Finding the Domain of the Rational Function We can factor the quadratic in the denominator: x 2 + 2 x − 8 = ( x + 4 ) ( x − 2 ) . Thus, the denominator is zero when x = − 4 or x = 2 . Since the rational function is only defined for x < 4 , we need to exclude − 4 and 2 from the interval ( − ∞ , 4 ) . Therefore, the domain of the first piece is ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , 4 ) .

Analyzing the Logarithmic Function The second piece is a logarithmic function: g ( x ) = lo g 3 ​ ( x + 5 ) for x ≥ 4 . Logarithmic functions are defined only for positive arguments. So, we need 0"> x + 5 > 0 , which means -5"> x > − 5 . Since this piece is defined for x ≥ 4 , we need to consider the intersection of -5"> x > − 5 and x ≥ 4 , which is simply x ≥ 4 . Therefore, the domain of the second piece is [ 4 , ∞ ) .

Combining the Domains Now, we combine the domains of the two pieces. The domain of the first piece is ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , 4 ) , and the domain of the second piece is [ 4 , ∞ ) . The union of these two domains is ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , 4 ) ∪ [ 4 , ∞ ) . This simplifies to ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , ∞ ) .

Final Answer Therefore, the domain of the piecewise function is ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , ∞ ) .


Examples
Understanding the domain of a piecewise function is crucial in many real-world applications, such as modeling the cost of electricity based on usage. For example, the price per kilowatt-hour (kWh) might be different for the first 500 kWh used compared to any additional kWh. Determining the domain helps ensure that the model is only applied to valid input values, preventing incorrect or nonsensical results. In this case, the domain represents the range of possible electricity usages for which the cost model is applicable.

Answered by GinnyAnswer | 2025-07-03

The domain of the piecewise function g ( x ) is ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , ∞ ) . This is derived from analyzing the rational part, which has restrictions at − 4 and 2 , and the logarithmic part, which is defined for x ≥ 4 . The correct choice from the options provided is B, ( − ∞ , − 4 ) ∪ ( − 4 , 2 ) ∪ ( 2 , ∞ ) .
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Answered by Anonymous | 2025-07-04