# Exponential functionHTML5

## Summary

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) = qou 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 → −∞ et f(x) → 0 when x → +∞ .
• q > 1 : The function is strictly increasing. f(x) → 0 when x → −∞. et 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 ou  f(x) = exp(x).

e is an irrational number, called a Euler number or Napier number: e ≈ 2,718  281 : e = f(1).

## Learning goals

• Analyze exponential functions with base q.
• Introduce the variable e.
• Study the influence of the parameters A and k in the general expression f(x) = Aq(kx).