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.
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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.