Let be a real valued function defined by` f(x) =(e^x-e^(|x|))/(e^x+e^|x|), ` then the range of `f(x)` is :
(A) R
(B) [0, 1]
(C) (0, 1)
(D) [0,0.5)

Explanation

To find the range of the function \( f(x) = \frac{e^x - e^{|x|}}{e^x + e^{|x|}} \), we will analyze it for two cases based on the definition of the absolute value function.

### Step 1: Analyze for \( x \geq 0 \)

When \( x \geq 0 \), we have \( |x| = x \). Therefore, the function simplifies to:

\[
f(x) = \frac{e^x - e^x}{e^x + e^x} = \frac{0}{2e^x} = 0
...