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Let $$f: (0, \infty) \rightarrow R$$ be given by $$f(x) = \int_{\frac{1}{x}}^{x} e^{-(t + \frac{1}{t})} \frac {dt}{t}.$$Then
f(x) is monotonically increasing on $$[1, \infty)$$
f(x) is monotonically decreasing on (0, 1)
$$f(x) + f \left(\frac{1}{x}\right) = 0,$$ for all $$x \in (0, \infty)$$
$$f(2^x)$$ is an odd function of x on R
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