Question

Determine the given function \(f\)to be an odd function.

Short answer

The resultant answer is odd because \(f( - x) = - f(x)\).

Step-by-step solution

Given data

The given series is \(f(x) = \frac{{1 - {e^{\frac{1}{x}}}}}{{1 + {e^{\frac{1}{x}}}}}\).

Concept of limits

Limits concept is based on the idea of closeness, used primarily to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values.

Obtain the values

Plug in \( - x\) as shown below.

\(\begin{array}{c}f( - x) = \frac{{1 - {e^{1/( - x)}}}}{{1 + {e^{1/( - x)}}}}\\f( - x) = \frac{{1 - \frac{1}{{{e^{1/x}}}}}}{{1 + \frac{1}{{{e^{1/x}}}}}}\end{array}\)

Multiply top and bottom by \({e^{1/x}}\) as follows:

\(\begin{array}{c}f( - x) = \frac{{1 - \frac{1}{{{e^{1/x}}}}}}{{1 + \frac{1}{{{e^{1/x}}}}}} \cdot \frac{{{e^{1/x}}}}{{{e^{1/x}}}}\\f( - x) = \frac{{{e^{1/x}} - 1}}{{{e^{1/x}} + 1}}\\f( - x) = \frac{{ - \left( {1 - {e^{1/x}}} \right)}}{{1 + {e^{1/x}}}}\\f( - x) = - \frac{{1 - {e^x}}}{{1 + {e^x}}}\end{array}\)

Because the function \(f( - x) = - f(x)\) is odd.