Functions
If f(x) = log(1 + x)/(1 - x), then f(2x/(1 + x²)) equals
Select the correct option:
Solution
2f(x)
- Substitution: Let u=1+x22x. We need to find f(u)=log(1−u1+u).
- Simplify the Argument:
- 1−u1+u=1−1+x22x1+1+x22x=1+x21+x2−2x1+x21+x2+2x
- =1+x2−2x1+x2+2x=(1−x)2(1+x)2=(1−x1+x)2
- Log Property: f(u)=log((1−x1+x)2)=2log(1−x1+x).
- Conclusion: This is exactly 2f(x).
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- functions
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
2f(x)
- Substitution: Let u=1+x22x. We need to find f(u)=log(1−u1+u).
- Simplify the Argument:
- 1−u1+u=1−1+x22x1+1+x22x=1+x21+x2−2x1+x21+x2+2x
- =1+x2−2x1+x2+2x=(1−x)2(1+x)2=(1−x1+x)2
- Log Property: f(u)=log((1−x1+x)2)=2log(1−x1+x).
- Conclusion: This is exactly 2f(x).
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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