Exponential Function

EXPONENTIAL FUNCTION

DEFINITION AND FORM

An exponential function is a function of the form:
f(x) = aˣ,
where 'a' is a positive real number different from 1 (a > 0, a ≠ 1), and x is any real number. The base 'a' determines whether the function is increasing or decreasing, while the variable x is the exponent.

The exponential function is the inverse function of the logarithmic function, which means:
if f(x) = aˣ, then f⁻¹(x) = logₐ(x).

PROPERTIES OF THE FUNCTION

• DOMAIN: Df = ℝ (the function is defined for all real numbers).
• RANGE: Zf = (0, ∞) (the function is always positive).
• The function never takes the value 0 or any negative values.
• The graph of the function never intersects the x-axis but approaches it – thus, it has a horizontal asymptote at y = 0.

INCREASING AND DECREASING

• If a > 1, the function is increasing.
• If 0 < a < 1, the function is decreasing.

Examples:

• f(x) = 2ˣ → an exponential function with base 2, increasing.
• f(x) = (1/2)ˣ → a decreasing function because the base is less than 1.

SOME CHARACTERISTIC PROPERTIES

• f(0) = a⁰ = 1, for any permissible base 'a'.
• f(1) = a¹ = a.
• The function is always positive.
• The function is continuous and differentiable over the entire set of real numbers (ℝ).
• The graph of the function does not intersect the x-axis, it only approaches it (y = 0 is an asymptote).

CALCULATION EXAMPLE

Let f(x) = 3ˣ.

• f(–2) = 3⁻² = 1/9
• f(0) = 1
• f(2) = 9
  The function increases rapidly, and its values are always positive.

EXPONENTIAL FUNCTION WITH BASE e

A special case is the function:
f(x) = eˣ, where e ≈ 2.718 (the base of the natural logarithm). This function plays an important role in higher mathematics, as its derivative is equal to itself.

CONCLUSION

The exponential function is fundamental for describing growth, decay, and changes in mathematics. Its form depends on the base, which determines whether it is increasing or decreasing. Due to its continuity, positivity, and simple properties, it is a key building block of mathematical analysis and the treatment of inverse functions.