Which of the following describes the domain of the piecewise function [tex]g(x)=\left\{\begin{array}{lll}\frac{x^2+3 x}{x^2+x-6} & \text { for } & x\ \textless \ 3 \\ \log _2(x+5) & \text { for } & x \geq 3\end{array}\right.[/tex] ?
A. [tex](-\infty, \infty)[/tex]
B. [tex](-\infty, 2) \cup(2, \infty)[/tex]
C. [tex](-\infty, 2) \cup(2,3) \cup(3, \infty)[/tex]
D. [tex](-\infty,-3) \cup(-3,2) \cup(2, \infty)[/tex]

Find the domain of the first piece x 2 + x − 6 x 2 + 3 x ​ for x < 3 by excluding the roots of the denominator x 2 + x − 6 = 0 , which are x = − 3 and x = 2 . The domain is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) .
Find the domain of the second piece lo g 2 ​ ( x + 5 ) for x ≥ 3 by ensuring 0"> x + 5 > 0 , which means -5"> x > − 5 . Combined with x ≥ 3 , the domain is [ 3 , ∞ ) .
Combine the domains of both pieces: ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) ∪ [ 3 , ∞ ) .
Simplify the combined domain to get the final answer: ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) ​ .

Explanation

Analyze the piecewise function We are given a piecewise function and asked to find its domain. The function is defined as

g ( x ) = { x 2 + x − 6 x 2 + 3 x ​ lo g 2 ​ ( x + 5 ) ​ for for ​ x < 3 x ≥ 3 ​
We need to consider the domain of each piece separately.

Find the domain of the first piece For the first piece, g ( x ) = x 2 + x − 6 x 2 + 3 x ​ for x < 3 . The domain is restricted by the denominator, which cannot be zero. So we need to find the values of x for which x 2 + x − 6 = 0 .

We can factor the quadratic as ( x + 3 ) ( x − 2 ) = 0 . Thus, the roots are x = − 3 and x = 2 . Since the first piece is defined for x < 3 , we must exclude x = − 3 and x = 2 from this interval. Therefore, the domain of the first piece is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) .

Find the domain of the second piece For the second piece, g ( x ) = lo g 2 ​ ( x + 5 ) for x ≥ 3 . The domain of a logarithm function is where the argument is greater than zero. So we need 0"> x + 5 > 0 , which means -5"> x > − 5 . Since the second piece is defined for x ≥ 3 , the domain is restricted to x ≥ 3 . Thus, the domain of the second piece is [ 3 , ∞ ) .

Combine the domains Now, we combine the domains of the two pieces. The first piece has domain ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) , and the second piece has domain [ 3 , ∞ ) . Combining these, we get ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) ∪ [ 3 , ∞ ) . This simplifies to ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .

Final Answer Therefore, the domain of the piecewise function is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .


Examples
Understanding the domain of a piecewise function is crucial in many real-world applications, such as modeling different pricing strategies for a product based on the quantity purchased. For example, a store might offer a lower price per item if a customer buys more than a certain quantity. This can be represented as a piecewise function, where the price per item changes at specific quantity thresholds. Determining the domain ensures that the function is valid for all possible purchase quantities, allowing for accurate modeling and analysis of the pricing strategy.

Answered by GinnyAnswer | 2025-07-03

The domain of the piecewise function is found by analyzing both pieces separately. For the first piece, we exclude values that make the denominator zero and find that the domain is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) . For the second piece, the domain is [ 3 , ∞ ) , leading to the final combined domain of ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .
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Answered by Anonymous | 2025-07-04