TS EAMCET · Maths · Functions

Let $ℝ$ be the set of all real numbers. Statement I : The function $f : (- \frac{π}{2}, \frac{π}{2}) \to ℝ$ defined by $f (x) = \sec x + \tan x$ is a one - one function. Statement II : The function $f : [0, \infty) \to ℝ$ defined by $f (x) = x^{2}$ is a one-one function. Which of the above statements is(are) true?

A Statement I is true, but Statement II is false
B Statement II is true, but Statement I is false
C Both Statement I and Statement II are true
D Both Statement I and Statement II are false

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Answer & Solution

Correct Answer

(C) Both Statement I and Statement II are true

Step-by-step Solution

Detailed explanation

Given, ℝ be the set of all real numbers. Now solving statement I : The function f:-π2,π2→ℝ defined by fx=secx+tanx Now differentiating the function with respect to x we get, f'x=secxsecx+tanx Now we know that secx is always positive in given domain…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let ℝ be the set of all real numbers. Statement I : The function f:-π2,π2→ℝ defined by fx=secx+tanx is a one - one function. Statement II : The function f:[0,∞)→ℝ defined by fx=x2 is a one-one function. Which of the above statements is(are) true?

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