Exponential Functions

Definition of Exponential Function

  • Exponential functions are formed by raising a constant base to a variable power, distinguished from algebraic functions where a variable is raised to a constant power.

  • If a > 0 and a≠1a \neq 1, the exponential function with base aa is defined as f(x)=axf(x) = a^x.

Properties of Exponents

  • Let aa and bb be positive real numbers, and let xx and yy be real numbers:

    • 1. a0=1a^0 = 1

    • 2. axay=ax+ya^x a^y = a^{x+y}

    • 3. axay=ax−y\frac{a^x}{a^y} = a^{x-y}

    • 4. (ax)y=axy(a^x)^y = a^{xy}

    • 5. (ab)x=axbx(ab)^x = a^x b^x

    • 6. (ab)x=axbx(\frac{a}{b})^x = \frac{a^x}{b^x}

    • 7. a−x=1axa^{-x} = \frac{1}{a^x}

Example 1 – Applying Properties of Exponents

  • a. (22)(23)=22+3=25=32(2^2)(2^3) = 2^{2+3} = 2^5 = 32

  • b. (22)(2−3)=22−3=2−1=12(2^2)(2^{-3}) = 2^{2-3} = 2^{-1} = \frac{1}{2}

  • c. (32)3=32(3)=36=729(3^2)^3 = 3^{2(3)} = 3^6 = 729

Graphs of Exponential Functions

  • The nature of exponential graphs can be determined via the point-plotting method or graphing utilities.

  • Behavior based on base aa:

    • Functions in the form y=axy = a^x or y=a−xy = a^{-x} exhibit specific growth or decay characteristics.

    • Graphs of axa^x (where a > 1) are strictly increasing.

    • Graphs of a−xa^{-x} (where a > 1) are strictly decreasing.

Example 5 – Analysis of f(x)=3−x−1f(x) = 3^{-x} - 1

  • A table of values identifies key coordinates:

    • (−2,8)(-2, 8), (−1,2)(-1, 2), (0,0)(0, 0), (1,−23)(1, -\frac{2}{3}), and (2,−89)(2, -\frac{8}{9}).

  • The horizontal asymptote of the graph is identified as y=−1y = -1.