Exponential quadratic equation and restricted domain for an inverse

Solve $e^{2x} - 8e^{x} + 7 = 0$ for $x$.

Let $g(x) = e^{2x} - 8e^{x} + 7$, where $x \in R$.

The function $g(x)$ has exactly one stationary point, a local minimum.

Find the largest value of $a$ such that when $g$ is restricted to the domain $(-\infty, a]$ it has an inverse function.