This page presents a detailed study of exponential functions using tables of values and graphs. Key properties such as domain, range, horizontal asymptotes, and intercepts are discussed thoroughly, along with the conditions governing exponential growth and decay.
Definition of the Exponential Function
The basic exponential function is defined by:
\[ f(x) = B^x \]where the base \( B \) satisfies \( B > 0 \) and \( B \neq 1 \). The domain of the exponential function \( f \) is the set of all real numbers (\(-\infty, +\infty\)).
Example 1: Exponential Functions with Base Greater Than 1
The table below shows values for exponential functions with bases greater than 1: \( 2^x \), \( 4^x \), and \( 7^x \).
| \( x \) | \( 2^x \) | \( 4^x \) | \( 7^x \) |
|---|---|---|---|
| \(-10\) | \(0.00097\) | \(9.53674 \times 10^{-7}\) | \(3.54013 \times 10^{-9}\) |
| \(-5\) | \(0.03125\) | \(0.00097\) | \(0.00006\) |
| \(-1\) | \(0.5\) | \(0.25\) | \(0.14285\) |
| \(0\) | \(1\) | \(1\) | \(1\) |
| \(1\) | \(2\) | \(4\) | \(7\) |
| \(3\) | \(8\) | \(64\) | \(343\) |
| \(5\) | \(32\) | \(1024\) | \(16807\) |
| \(10\) | \(1024\) | \(1048576\) | \(282475249\) |
Example 2: Exponential Functions with Base Between 0 and 1
The table below shows values for exponential functions with bases between 0 and 1: \( 0.2^x \), \( 0.5^x \), and \( 0.8^x \).
| \( x \) | \( 0.2^x \) | \( 0.5^x \) | \( 0.8^x \) |
|---|---|---|---|
| \(-10\) | \(9765625\) | \(1024\) | \(9.31323\) |
| \(-5\) | \(3125\) | \(32\) | \(3.05175\) |
| \(-1\) | \(5\) | \(2\) | \(1.25\) |
| \(0\) | \(1\) | \(1\) | \(1\) |
| \(1\) | \(0.2\) | \(0.5\) | \(0.8\) |
| \(3\) | \(0.008\) | \(0.125\) | \(0.512\) |
| \(5\) | \(0.00032\) | \(0.03125\) | \(0.32768\) |
| \(10\) | \(1.024 \times 10^{-7}\) | \(0.0009765625\) | \(0.1073741824\) |
Properties of Exponential Functions
Based on the analysis of tables and graphs, the key properties of \( f(x) = B^x \) include:
- Domain: The domain of \( f(x) = B^x \) is all real numbers (\(-\infty, +\infty\)).
- Exponential Growth (\( B > 1 \)): If \( B > 1 \), the function is strictly increasing: \[ \lim_{x \to +\infty} B^x = +\infty, \qquad \lim_{x \to -\infty} B^x = 0 \]
- Exponential Decay (\( 0 < B < 1 \)): If \( 0 < B < 1 \), the function is strictly decreasing: \[ \lim_{x \to +\infty} B^x = 0, \qquad \lim_{x \to -\infty} B^x = +\infty \]
- Horizontal Asymptote: The line \( y = 0 \) serves as a horizontal asymptote for all basic exponential functions.
- Intercepts: All exponential functions satisfy \( f(0) = 1 \), yielding a \(y\)-intercept at \( (0,1) \), and have no \(x\)-intercepts.
- Range: The range of \( f(x) = B^x \) is the open interval \( (0, +\infty) \).
Writing Exponentials Using the Natural Base
Let \( f(x) = B^x \). Since \( B = e^{\ln B} \), any exponential function can be rewritten using the natural base \( e \):
\[ B^x = (e^{\ln B})^x = e^{x \ln B} \]Thus,
\[ \boxed{B^x = e^{x \ln B}} \]In calculus and advanced mathematics, natural exponentials are standard because their derivatives and integrals have exceptionally simple forms.