iven below are two statements : Statement I : The function defined by is one-one. Statement II : The function defined by is many-one. In the light of the above statements, choose the correct answer from the options given below :

Question

Given below are two statements :

Statement I : The function $f : ℝ \to ℝ$ defined by $f (x) = \frac{x}{1 + | x |}$ is one-one.

Statement II : The function $f : ℝ \to ℝ$ defined by $f (x) = \frac{x^{2} + 4 x - 30}{x^{2} - 8 x + 18}$ is many-one.

In the light of the above statements, choose the correct answer from the options given below : [2026]

Smart Achievers · Accepted Answer

(4)

$Statement 1 :$ $f (x)$ $f (x) = \frac{x}{1 + | x |}$

$f (x)$ $=$ $f : ℝ \to ℝ$

$f(x) is one-one$

$Statement 2 :$ $f (x)$ $= \frac{x^{2} + 4 x - 30}{x^{2} - 8 x + 18}, f (0) = \frac{- 30}{18} = \frac{- 5}{3}$

$\frac{- 5}{3} = \frac{x^{2} + 4 x - 30}{x^{2} - 8 x + 18}$

$On solving x = 0, - 1$

$\Rightarrow f (0) = f (- 1) = \frac{- 5}{3}$

$∴$ $f (x) is many-one$

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