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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) = q^{x }or f(x) = exp_{q}(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 exp_{q}(x) is a convex function that passes through the coordinates (0, 1) : ∀ q, q^{0} = 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) = e^{x} or f(x) = exp(x).

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

- To analyze exponential functions with base q.
- To introduce the exponential constant e.
- To study the influence of the parameters A and k in the general expression f(x) = Aq
^{(kx)}.

It is interesting to compare exponential functions with a known increasing function such as x^{n}. An exponential function will always increase faster than a power function. Move the cursor on each…