To determine whether the function [tex]\( g(x) = 4 \cdot 5^x \)[/tex] represents exponential growth or decay, follow these steps:
1. Understand the Exponential Function Form:
The general form of an exponential function is [tex]\( g(x) = a \cdot b^x \)[/tex], where:
- [tex]\( a \)[/tex] is a constant multiplier (initial value or coefficient).
- [tex]\( b \)[/tex] is the base of the exponential term.
- [tex]\( x \)[/tex] is the exponent.
2. Identify Key Components:
For the given function [tex]\( g(x) = 4 \cdot 5^x \)[/tex]:
- The constant multiplier [tex]\( a \)[/tex] is 4.
- The base [tex]\( b \)[/tex] is 5.
3. Determine Growth or Decay:
- If the base [tex]\( b \)[/tex] is greater than 1, the function represents exponential growth.
- If the base [tex]\( b \)[/tex] is between 0 and 1 (0 < b < 1), the function represents exponential decay.
4. Analyze the Base:
Here, the base [tex]\( b \)[/tex] is 5, which is greater than 1.
Therefore, [tex]\( g(x) = 4 \cdot 5^x \)[/tex] represents exponential growth because the base equals 5.
So, the correct statement is:
The function represents exponential growth because the base equals 5.
