ExpFx

ExpFx is a notational convention used in some mathematical and programming contexts for the exponential function. In this article, ExpFx(x) is defined as e^x, where e is the base of natural logarithms. It is equivalent to the common exp function often written as exp(x).

Properties of ExpFx include that its domain is all real numbers and its range is (0, ∞). The function is positive, strictly increasing, and convex. The derivative equals the function itself: d/dx ExpFx(x) = ExpFx(x). The second derivative is also ExpFx(x). The indefinite integral is ExpFx(x) + C. The Taylor series is ExpFx(x) = sum from n = 0 to infinity of x^n / n!. It is the inverse of the natural logarithm on positive values: ln(ExpFx(x)) = x for all real x, and ExpFx(ln y) = y for y > 0.

Key identities include ExpFx(x+y) = ExpFx(x) · ExpFx(y); ExpFx(x) · ExpFx(y) = ExpFx(x+y).

Generalizations cover scaling and translation, yielding expressions such as a · ExpFx(bx + c) for constants a > 0. More broadly, exponentials extend to higher dimensions via matrix or vector exponentials in appropriate contexts.

Applications of ExpFx span modeling continuous growth or decay (population dynamics, bacterial growth, radioactive decay), compound interest, differential equations, and probability distributions (such as the log-normal distribution). It also appears in signal processing as a fundamental building block for filters and envelopes.

Computational notes: in many programming languages ExpFx is implemented as exp or Math.exp; some libraries expose exp under different namespaces. In formula writing, ExpFx(x) is interchangeable with e^x.