First slide

Functions (XII)

Question

$The range of the function f (x) = \frac{e^{x} - e^{| x |}}{e^{x} + e^{| x |}} is$

Moderate

A

$(- \infty, \infty)$
B

$[0, 1)$
C

$(- 1, 0]$
D

$(- 1, 1)$

Solution

$f (x) = \frac{e^{x} - e^{| x |}}{e^{x} + e^{| x |}} = \{\begin{cases} 0, x \geq 0 \\ \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}, x < 0 \end{cases}$
$Clearly, f (x) is identically zero if x \geq 0 - - - - (1)$
$\begin{array}{ll} ∵ x < 0 \\ e^{2 x} < 1 or 0 < e^{2 x} < 1 \\ ∴ 0 < \frac{1 + y}{1 - y} < 1 \end{array}$
$\begin{array}{l} or \frac{1 + y}{1 - y} > 0 and \frac{1 + y}{1 - y} < 1 \\ or (y + 1) (y - 1) < 0 and \frac{2 y}{1 - y} < 0 \end{array}$
$i.e., - 1 < y < 1 and y < 0 or y > 1$
$or - 1 < y < 0 - - - - - - (2)$
$Combining (1) and (2), we get - 1 < y \leq 0 or Range = (- 1, 0] .$

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