Question

Given below are two statements: Statement I: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one-one. Statement II: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many-one. In the light of the above statements, choose the correct answer.

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

Hint:

A function defined on \(\mathbb{R}\) is one-one if it is strictly monotonic throughout its domain.

Answer 1

The Correct Option is B

Statement I: For \(x\ge 0\), \[ f(x)=\frac{x}{1+x}, \quad \text{which is strictly increasing.} \] For \(x<0\), \[ f(x)=\frac{x}{1-x}, \quad \text{which is also strictly increasing.} \] Moreover, \[ \lim_{x\to-\infty} f(x)=-1, \qquad \lim_{x\to\infty} f(x)=1. \] Since the function is strictly increasing on \((-\infty,\infty)\), it is one-one. \[ \Rightarrow \text{Statement I is true.} \] Statement II: The given rational function has quadratic numerator and denominator. Such functions generally take the same value for more than one \(x\), unless they are strictly monotonic on their domain. Indeed, by symmetry and algebraic inspection, one can find distinct values of \(x\) giving the same function value. \[ \Rightarrow \text{The function is many-one.} \] \[ \Rightarrow \text{Statement II is true.} \] Final Conclusion: \[ \boxed{\text{Both Statement I and Statement II are true}} \]