Functions
The range of the function f(x) = x/(1 + |x|) is
Select the correct option:
Solution
(-1, 1)
- Piecewise Definition:
- For x≥0, f(x)=1+xx. As x→∞,f(x)→1. Range =[0,1).
- For x<0, f(x)=1−xx. As x→−∞,f(x)→−1. Range =(−1,0).
- Combine Domains: The total range is the union of (−1,0)∪[0,1)=(−1,1).
- Boundaries Analysis: Note that 1+∣x∣x can never exactly equal 1 because ∣x∣<1+∣x∣ for all finite x. Similarly for −1. Thus, the interval is open.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- functions
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
(-1, 1)
- Piecewise Definition:
- For x≥0, f(x)=1+xx. As x→∞,f(x)→1. Range =[0,1).
- For x<0, f(x)=1−xx. As x→−∞,f(x)→−1. Range =(−1,0).
- Combine Domains: The total range is the union of (−1,0)∪[0,1)=(−1,1).
- Boundaries Analysis: Note that 1+∣x∣x can never exactly equal 1 because ∣x∣<1+∣x∣ for all finite x. Similarly for −1. Thus, the interval is open.
This hard difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of functions. It appeared in the 2025 exam.
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