The exponential function is the inverse of the logarithmic function. Therefore, its domain and range are inverted compared to the exponential function. An exponential function is defined as a function of the form:

\[ y = e^x \quad \text{with} \quad x : \forall x \in \mathbb{R}\]

  • Domain: \(\mathbb{R} \)
  • Range: \(y \in \mathbb{R} : y > 0\)

The graphical representation of the function is:

Graphical representation
of the function \(f(x) = e^x\).

Graphical representation
of the function \(f(x) = e^{-x}\).


  • Limit: \[\lim_{{x \to -\infty}} e^x = 0\]

  • Derivative: \[\frac{{d}}{{dx}} e^x = e^x\]

  • Indefinite integral: \[\int e^x \ dx = e^x + c\]

  • Definite integral \[\int_{-\infty}^{0} e^x \ dx = 1\]