Find the domain and range of the function $\frac{-2 x(x+3)^2(x+2)}{3 x(x+2)(x+3)^2}$.
A. $D:(-\infty,-3) \cup(-3,-2) \cup(-2,0) \cup(0, \infty) R:(-\infty, 0) \cup(0, \infty)$
B. $D:(-\infty,-3) \cup(-3,-2) \cup(-2, \infty) R:(-\infty, 0) \cup(0, \infty)$
C. $D:(-\infty,-3) \cup(-3,-2) \cup(-2, \infty) R:(-\infty, 2 / 3) \cup(2 / 3, \infty)$
D. $D:(-\infty,-3) \cup(-3,-2) \cup(-2,0) \cup(0, \infty) R:(-\infty,-2 / 3) \cup(-2 / 3, \infty)$
Answer (2)
Simplify the function and identify values where the denominator is zero.
Determine the domain by excluding these values: ( − ∞ , − 3 ) ∪ ( − 3 , − 2 ) ∪ ( − 2 , 0 ) ∪ ( 0 , ∞ ) .
Find the simplified function's value, which gives the range: { − 3 2 } .
The closest answer among the options is D, but the range is not correctly represented. The correct range is just the single value { − 3 2 } .
Explanation
Understanding the Problem We are given the function f ( x ) = 3 x ( x + 2 ) ( x + 3 ) 2 − 2 x ( x + 3 ) 2 ( x + 2 ) . We need to find its domain and range. The domain consists of all real numbers for which the function is defined, and the range is the set of all possible output values of the function.
Simplifying the Function First, we simplify the function by canceling common factors in the numerator and the denominator. We have f ( x ) = 3 x ( x + 2 ) ( x + 3 ) 2 − 2 x ( x + 3 ) 2 ( x + 2 ) = 3 − 2 , provided that x = 0 , x = − 2 , and x = − 3 .
Finding the Domain The original denominator is 3 x ( x + 2 ) ( x + 3 ) 2 . The function is undefined when the denominator is equal to zero. Thus, we need to find the values of x for which 3 x ( x + 2 ) ( x + 3 ) 2 = 0. This occurs when x = 0 , x = − 2 , or x = − 3 . Therefore, the domain of the function is all real numbers except 0 , − 2 , and − 3 . In interval notation, the domain is ( − ∞ , − 3 ) ∪ ( − 3 , − 2 ) ∪ ( − 2 , 0 ) ∪ ( 0 , ∞ ) .
Finding the Range Since the simplified function is f ( x ) = − 3 2 for all x in the domain, the range is just the single value { 3 − 2 } .
Comparing with Options Comparing our results with the given options, we see that the domain is ( − ∞ , − 3 ) ∪ ( − 3 , − 2 ) ∪ ( − 2 , 0 ) ∪ ( 0 , ∞ ) and the range is { − 3 2 } . However, none of the options exactly match this. Option D has the correct domain, but the range is incorrect. It seems there might be a typo in the options. The correct range should be just the single value − 3 2 . Since the domain in option D is correct, we can consider that the range should have been R = { − 3 2 } . However, since this is not an option, and the closest one is D, we will analyze it. Option D states R : ( − ∞ , − 2/3 ) ∪ ( − 2/3 , ∞ ) , which is incorrect. The range is just the single value − 3 2 .
Final Answer The domain of the function is ( − ∞ , − 3 ) ∪ ( − 3 , − 2 ) ∪ ( − 2 , 0 ) ∪ ( 0 , ∞ ) , and the range is { − 3 2 } . Among the given options, option D has the correct domain, but the range is not correct. However, it is the closest to the correct answer. There might be a typo in the options, and the range should have been just the single value − 3 2 .
Examples
Consider a scenario where you are analyzing the efficiency of a manufacturing process. The function represents the ratio of output to input, but there are certain input levels (x-values) where the machinery breaks down (x = -3, -2, 0). The domain tells you the safe operating input levels, and the range tells you the constant efficiency level when the machinery is operating correctly. This helps in optimizing the process by avoiding breakdown points and understanding the achievable efficiency.
The domain of the function is ( − ∞ , − 3 ) ∪ ( − 3 , − 2 ) ∪ ( − 2 , 0 ) ∪ ( 0 , ∞ ) , while the range is the single value { -\frac{2}{3} }. Among the options provided, option D correctly states the domain but incorrectly lists the range. Therefore, although option D is the closest, the correct range needs to reflect just the single value { -\frac{2}{3} }.
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