What is the range of [tex]f(x)=\frac{-2}{x-2}-2[/tex]?
A. [tex](-\infty, 0) \cup(0, \infty)[/tex]
B. [tex](-\infty, 2) \cup(2, \infty)[/tex]
C. [tex](-\infty,-2) \cup(2, \infty)[/tex]
D. [tex](-\infty,-2) \cup(-2, \infty)[/tex]
Answer (2)
Identify the vertical asymptote at x = 2 .
Observe the horizontal asymptote at y = − 2 .
Note that the function can take on any value except − 2 .
The range of the function is ( − in f t y , − 2 ) c u p ( − 2 , in f t y ) .
Explanation
Understanding the Problem We are given the function f ( x ) = f r a c − 2 x − 2 − 2 and asked to find its range. The range is the set of all possible output values of the function.
Analyzing the Function To find the range, we can analyze the function's behavior. The function has a vertical asymptote at x = 2 , where the denominator x − 2 is zero. As x approaches 2 from the left, x − 2 is a small negative number, so f r a c − 2 x − 2 is a large positive number. As x approaches 2 from the right, x − 2 is a small positive number, so f r a c − 2 x − 2 is a large negative number. Therefore, the function can take on very large positive and negative values.
Finding Asymptotes The function also has a horizontal asymptote. As x approaches p m in f t y , the term f r a c − 2 x − 2 approaches 0, so f ( x ) approaches − 2 . However, f ( x ) never actually equals − 2 , because f r a c − 2 x − 2 is never exactly 0.
Determining the Range Therefore, the range of the function is all real numbers except − 2 . In interval notation, this is ( − in f t y , − 2 ) c u p ( − 2 , in f t y ) .
Final Answer The range of f ( x ) = f r a c − 2 x − 2 − 2 is ( − in f t y , − 2 ) c u p ( − 2 , in f t y ) .
Examples
Understanding the range of a function is crucial in many real-world applications. For example, if f ( x ) represents the profit of a company based on the number of units produced ( x ), knowing the range tells you the possible profit values the company can achieve. If the range is ( − in f t y , − 2 ) c u p ( − 2 , in f t y ) , it means the company can make any profit except exactly -2 (which could represent a loss of $2). This helps in setting realistic expectations and making informed business decisions.
The range of the function f ( x ) = x − 2 − 2 − 2 is all real numbers except − 2 . In interval notation, this is expressed as ( − ∞ , − 2 ) ∪ ( − 2 , ∞ ) . Therefore, the selected option is D: ( − ∞ , − 2 ) ∪ ( − 2 , ∞ ) .
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