1. The problem is to understand the function $f(x) = e^i$. 2. Here, $e$ is the base of the natural logarithm, approximately 2.71828, and $i$ is the imaginary unit, defined by $i^2 = -1$. 3. The function $f(x) = e^i$ is a constant complex number because the exponent is a constant $i$, not depending on $x$. 4. Using Euler's formula, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, for any real number $\theta$. 5. In this case, $\theta = 1$, so $e^i = \cos(1) + i\sin(1)$. 6. Evaluating numerically, $\cos(1) \approx 0.5403$ and $\sin(1) \approx 0.8415$. 7. Therefore, $f(x) = e^i \approx 0.5403 + 0.8415i$. 8. This is a complex constant, not a function of $x$. Final answer: $$f(x) = e^i = \cos(1) + i\sin(1) \approx 0.5403 + 0.8415i$$