Functions
If f(x) = x/(x² + 1), then the range of f is
Select the correct option:
Solution
[-1/2, 1/2]
- Method 1 (Algebraic): Let y=x2+1x⟹yx2−x+y=0.
- For x to be real, the discriminant D of this quadratic in x must be ≥0.
- D=(−1)2−4(y)(y)=1−4y2≥0.
- 4y2≤1⟹y2≤1/4⟹−1/2≤y≤1/2.
- Method 2 (Inequality): Use AM≥GM for x>0: x+1/x≥2⟹x+1/x1≤1/2.
- Result: Range is [−1/2,1/2].
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- functions
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
[-1/2, 1/2]
- Method 1 (Algebraic): Let y=x2+1x⟹yx2−x+y=0.
- For x to be real, the discriminant D of this quadratic in x must be ≥0.
- D=(−1)2−4(y)(y)=1−4y2≥0.
- 4y2≤1⟹y2≤1/4⟹−1/2≤y≤1/2.
- Method 2 (Inequality): Use AM≥GM for x>0: x+1/x≥2⟹x+1/x1≤1/2.
- Result: Range is [−1/2,1/2].
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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