f(x) = (e2x−1/e2x+1) is

The correct option is "an increasing function".

To determine the behavior of the function, let us first write down the given function clearly:
$$f\left(x\right)=\frac{e^{2x}-1}{e^{2x}+1}$$

We can simplify the expression for the function by dividing both the numerator and the denominator by $e^x$:
$$f\left(x\right)=\frac{\frac{e^{2x}}{e^x}-\frac{1}{e^x}}{\frac{e^{2x}}{e^x}+\frac{1}{e^x}}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$
This is the standard definition of the hyperbolic tangent function, $\tanh \left(x\right)$.

To analyze whether the function is increasing or decreasing, we find its first derivative with respect to x, denoted as $f\text{'}\left(x\right)$. Using the quotient rule:
$$f^{\prime }\left(x\right)=\frac{d}{dx}\left[\frac{e^{2x}-1}{e^{2x}+1}\right]$$

Applying the quotient rule ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$ where $u=e^{2x}-1$ and $v=e^{2x}+1$:
$$f^{\prime }\left(x\right)=\frac{\left(2e^{2x}\right)(e^{2x}+1)-(e^{2x}-1)\left(2e^{2x}\right)}{{(e^{2x}+1)}^2}$$

Factoring out the common term $2e^{2x}$ in the numerator:
$$f^{\prime }\left(x\right)=\frac{2e^{2x}[(e^{2x}+1)-(e^{2x}-1)]}{{(e^{2x}+1)}^2}$$

Simplifying the terms inside the square brackets:
$$e^{2x}+1-e^{2x}+1=2$$

Substitute this back into the derivative equation:
$$f^{\prime }\left(x\right)=\frac{2e^{2x}\cdot 2}{{(e^{2x}+1)}^2}=\frac{4e^{2x}}{{(e^{2x}+1)}^2}$$

Now, let us analyze the sign of $f^{\prime }\left(x\right)$ for all real values of x:
1. The exponential function $e^{2x}$ is strictly positive for all real values of x (i.e., $e^{2x}>0$).
2. Consequently, the numerator $4e^{2x}>0$ for all x.
3. The denominator is a squared term, ${(e^{2x}+1)}^2$, which is also strictly positive for all real values of x.

Since both the numerator and the denominator are strictly positive, we have:
$$f^{\prime }\left(x\right)>0\text{\hspace{0.17em}for all\hspace{0.17em}}x\in ℝ$$

Because the first derivative of the function is strictly positive everywhere on its domain, the function is an increasing function.