#### To determine

**To find:** To prove that the given function is an odd function.

#### Answer

the given function is an odd function

**You may know**:

A function *f* is said to be an odd function is for every point *x* in its domain, the following condition is satisfied:

f(−x)=−f(x)

Graphically, the graph of an odd function is symmetric with respect to the origin.

#### Explanation

**Given:**

The exponential function, f(x)=1−e1/x1+e1/x

**Calculation:**

The given function is f(x)=1−e1/x1+e1/x.

Find the values of the functions at the points ± 1, ± 2, ± 3.

** > **

** > **

** > **

** > **

** > **

** > **

The table of values is shown below:

*x* |
f(x)=1−e1/x1+e1/x |

1 |
–0.46211 |

−1 |
0.46211 |

2 |
–0.24491 |

−2 |
0.24491 |

3 |
–0.16514 |

−3 |
0.16514 |

From the above table, it can be noted that f(−x)=−f(x) is satisfied for the values ± 1, ± 2, ± 3.

Plot the graph of the given function using the above points.

The graph of the function is shown below:

The graph of the given function represents a rectangular hyperbola.

It has two asymptotes. The horizontal axis that is *x*-axis acts as the horizontal asymptote and the vertical axes that is *y*-axis acts as a vertical asymptote.

From the above graph, it can be noted that the graph is symmetric with respect to the origin.

This proves that the given function is an odd function.

**Conclusion:** the given function is an odd function.