Functions
If f: [0, ∞) → [0, ∞) is defined by f(x) = x/(1 + x), then f is
Select the correct option:
Solution
One-one but not onto
- One-one (Injection): f′(x)=(1+x)2(1+x)−x=(1+x)21. Since the derivative is strictly positive for x∈[0,∞), the function is strictly increasing and thus one-one.
- Range Analysis: As x→∞, f(x)→xx=1. Since x is always positive, f(x) is always less than 1. Thus, Range =[0,1).
- Onto (Surjection): The codomain is provided as [0,∞). Since the range ([0,1)) is a proper subset of the codomain, the function is not onto.
- Conclusion: One-one but not onto.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- functions
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
One-one but not onto
- One-one (Injection): f′(x)=(1+x)2(1+x)−x=(1+x)21. Since the derivative is strictly positive for x∈[0,∞), the function is strictly increasing and thus one-one.
- Range Analysis: As x→∞, f(x)→xx=1. Since x is always positive, f(x) is always less than 1. Thus, Range =[0,1).
- Onto (Surjection): The codomain is provided as [0,∞). Since the range ([0,1)) is a proper subset of the codomain, the function is not onto.
- Conclusion: One-one but not onto.
This medium 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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