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)= x²+4 x-30 x²-8 x+18 is many-one. In the light of the above statements, choose the correct answer from the options given below :

Step-by-step solution

Statement I: f(x) = x 1+|x| . For x 0 , f(x) = x 1+x is strictly increasing. For x 0 , also strictly increasing. The function is continuous at x = 0 , so f is strictly increasing on R and hence one-one. Statement I is true. Statement II: f(x) = x^2+4x-30 x^2-8x+18 . The denominator (x-4)^2 + 2 > 0 for all x . f'(x) = -12(x^2 - 8x + 14) (x^2 - 8x + 18)^2 f'(x) = 0 x = 4 2 . Since f' changes sign, f is not monotonic and hence many-one. Statement II is true. Both statements are true.