The range of the function `f(x)=(e^x-e^(|x|))/(e^x+e^(|x|))` is

Options

A. `(-oo,oo)`
B. `[0,1)`
C. `(-1,0]`
D. `(-1,1)`

Answer

C

Explanation

`f(x)=(e^(x)-e^(|x|))/(e^(x)+e^(|x|))={(0",", x ge 0),((e^(x)-e^(-x))/(e^(x)+e^(-x))",", x lt 0):}`
Clearly, `f(x)` is identically zero if `x ge 0`
If `x lt 0, " let " y=f(x)=(e^(x)-e^(-x))/(e^(x)+e^(-x)) " or " e^(2x)=(1+y)/(1-y) " (1)" `
` :' x lt 0`
` e^(2x) lt 1 " or " 0 lt e^(2x) lt 1`
` :. 0 lt (1+y)/(1-y) lt 1`
or `(1+y)/(1-y) gt 0 " and " (1+y)/(1-y) lt 1`
or `(1+y)(1-y) lt 0 " and " (2y)/(1-y) lt 0`
i.e., `-1 lt y lt 1 " and " y lt 0 " or " y gt 1`
or `-1 lt y lt 0 " (2)" `
Combining (1) and (2), we get `-1 lt y le 0` or Range `=(-1,0].`