Section 6.3: Exponential Functions
An exponential function has the form \( f(x) = a \cdot b^x \), where \( a \neq 0 \), \( b > 0 \), and \( b \neq 1 \). It models growth or decay.
Example 1
Evaluate \( f(x) = 2 \cdot 3^x \) for \( x = 4 \)
\( f(4) = 2 \cdot 3^4 = 2 \cdot 81 = 162 \)
Example 2
Identify if \( g(x) = 5 \cdot (1/2)^x \) represents growth or decay.
Since the base \( 1/2 < 1 \), this is exponential decay.
Example 3
Find \( h(3) \) for \( h(x) = 4 \cdot 2^x \)
\( h(3) = 4 \cdot 2^3 = 4 \cdot 8 = 32 \)
Practice Problems
- Evaluate \( f(x) = 3 \cdot 5^x \) for \( x = 2 \)
- Determine growth or decay for \( g(x) = 7 \cdot (3/4)^x \)
- Evaluate \( h(x) = 6 \cdot 2^x \) for \( x = 5 \)
- Write an exponential function for doubling every period starting at 2
- Write an exponential function for halving every period starting at 8