Consider the following statements: I. f(t) is an odd function. II. g(t) is an odd function. Which of the statements given above is/are correct?
Direction: Consider the following for the two (02) items that follow : Let f(t)=\ln(t+\sqrt{1+t^{2}}) and g(t)=\tan(f(t)).
- A. I only
- B. II only
- C. Both I and II ✓
- D. Neither I nor II
Correct Answer: C. Both I and II
Explanation
To check if f(t) is odd, evaluate f(-t) = \ln(-t+\sqrt{1+t^2}). Rationalizing the argument gives \ln(\frac{1}{t+\sqrt{1+t^2}}) = -\ln(t+\sqrt{1+t^2}) = -f(t). Thus, f(t) is odd. For g(t), g(-t) = \tan(f(-t)) = \tan(-f(t)) = -\tan(f(t)) = -g(t). Thus, g(t) is also odd. Both statements are correct.
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