Definition: ExponentialFunction

Definition: Exponential Function (OpenStax Textbook)

An exponential function is of the form \(b^x\) where \(b>0\) and \(b \neq 1\).

Note: An exponential function is not an algebraic function.

Key Characteristics of Exponential Function

• Domain is all real numbers
• Range is positive real numbers
• When \(b > 1\) the function is increasing on its entire domain.
  • As \(x\) decreases (approaches negative infinity) the output gets close to zero.
  • As \(x\) increases the output increases without bound.
  • These functions are concave up. (We will learn the precise meaning of concavity later this semester.)
• When \(0< b < 1\) the function is decreasing on its entire domain.
  • As \(x\) decreases (approaches negative infinity) the output grows without bound.
  • As \(x\) increases the output gets close to zero.
  • These functions are concave up. (We will learn the precise meaning of concavity later this semester.)

Properties of Exponents

If \(a > 0\), \(b >0\), and \(m\) and \(n\) are any real number, then - \(a^m \cdot a^n = a^{m+n}\) - \(\frac{a^m}{a^n} = a^{m-n}\) - \((a^m)^n = a^{mn}\) - \(\frac{1}{a^n} = a^{-n}\) - \(\sqrt[n]{a} = a^{1/n}\) - \(\sqrt[n]{a^m} = a^{m/n}\) - \(a^m \cdot b^m = (a \cdot b)^m\) - \(\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m\)