Exponential function

An exponential function is a power function where the variable x, is the exponent. In its most basic form, an exponential function is written as follows: f(x) = qor f(x) = expq(x).

The parameter q is the base of the exponent. It is a strictly positive real number not equal to 1.

The variation in an exponential function falls within 2 intervals:

• 0 < q < 1: The function is strictly decreasing:  f(x) → +∞ when x → −∞ and f(x) → 0 when x → +∞.
• q > 1: The function is strictly increasing. f(x) → 0 when x → −∞. and f(x) → +∞ when x → 0.

If q = 1, then the function is constant. It is equivalent to the equation y = 1.

If q ≠ 1, the exponential function has the same asymptote with the equation y = 0.

The exponential function expq(x) is a convex function that passes through the coordinates (0, 1) : ∀ q, q0 = 1.

Special Case: The exponential function with a tangent at the point (0,1) with line y = x, is the exponential function with the base e. It is written as f(x) = ex or  f(x) = exp(x).

e is an irrational number, called the exponential constant: e ≈ 2,718  281. e = f(1).