Question
Given below are two statements:
Statement I: The function $f:\mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x) = \frac{x}{1 + |x|}$ is one-one.
Statement II : The function $f:\mathbf{R} \rightarrow \mathbf{R}$ 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 from the options given below :
Options
Solution
Statement 1: $f(x) = \frac{x}{1 + |x|}$
$$f(x) = \left\{ \begin{matrix} \frac{x}{1 + x} & x \geq 0 \\ \frac{x}{1 - x} & x < 0 \end{matrix} \right.\ $$
$f(x)$ is one-one
Statement 2: $f(x) = \frac{x^{2} + 4x - 30}{x^{2} - 8x + 18},f(0) = \frac{- 30}{18} = \frac{- 5}{3}$
$$\frac{- 5}{3} = \frac{x^{2} + 4x - 30}{x^{2} - 8x + 18} $$On solving $x = 0, - 1$
$$\Rightarrow f(0) = f( - 1) = \frac{- 5}{3} $$$\therefore f(x)$ is many-one
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