First slide

Functions (XII)

Question

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

Moderate

A

$\begin{array}{l} (- \infty, \infty) \end{array}$
B

$\begin{array}{l} [0, 1) \end{array}$
C

$\begin{array}{l} (- 1, 0] \end{array}$
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)
If $x < 0, let y = f (x) = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} or e^{2 x} = \frac{1 + y}{1 - y}$
$\begin{array}{l} x < 0 \\ e^{2 x} < 1 or 0 < e^{2 x} < 1 \\ ∴ 0 < \frac{1 + y}{1 - y} < 1 \\ or \frac{1 + y}{1 - y} > 0 and \frac{1 + y}{1 - y} < 1 \\ or (y + 1) (y - 1) < 0 and \frac{2 y}{1 - y} < 0 \\ i . e ., - 1 < y < 1 and y < 0 or y > 1 \end{array}$
$or - 1 < y < 0$ (2)
Combining (1) and (2),we get $- 1 < y \leq 0$ or Range = ( -1, 0]

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