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Given below are two statements : Statement I : The function f : R → R defined by f(x) = x/(1+|x|) is one-one. Statement II : The function f : R → R defined by f(x) = (x^2 + 4x − 30)/(x^2 − 8x + 18) is many-one. In the light of the above statements, choose the correct answer from the options given below : (1) Both Statement I and Statement II are false. (2) Both Statement I and Statement II are true. (3) Statement I is false but Statement II is true. (4) Statement I is true but Statement II is false.

1. Both Statement I and Statement II are false.
2. Both Statement I and Statement II are true.
3. Statement I is false but Statement II is true.
4. Statement I is true but Statement II is false.

Correct answer: Both Statement I and Statement II are true.

Solution

Statement I is true because the function f(x) = x/(1+|x|) is strictly increasing, which ensures that it is one-one. Statement II is also true since the function f(x) = (x^2 + 4x − 30)/(x^2 − 8x + 18) can take the same value for different inputs, indicating it is many-one.

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