To identify which of the given exponential functions has a growth factor of 5, we need to examine the structure of each function closely.
### Understanding Exponential Functions
An exponential function is generally of the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where:
- [tex]\( a \)[/tex] is a constant term (also called the coefficient or initial value),
- [tex]\( b \)[/tex] is the base (also called the growth factor if [tex]\( b > 1 \)[/tex] or decay factor if [tex]\( 0 < b < 1 \)[/tex]),
- [tex]\( x \)[/tex] is the exponent.
### Given Exponential Functions
We are provided with two functions:
1. [tex]\( f(x) = 2 \cdot 5^x \)[/tex]
2. [tex]\( f(x) = 0.5 \cdot 2^x \)[/tex]
### Analyzing the Functions
Let's identify the growth factor in each function by focusing on the base [tex]\( b \)[/tex]:
1. For the function [tex]\( f(x) = 2 \cdot 5^x \)[/tex]:
- The base [tex]\( b \)[/tex] is [tex]\( 5 \)[/tex].
- Therefore, the growth factor for this function is [tex]\( 5 \)[/tex].
2. For the function [tex]\( f(x) = 0.5 \cdot 2^x \)[/tex]:
- The base [tex]\( b \)[/tex] is [tex]\( 2 \)[/tex].
- Consequently, the growth factor for this function is [tex]\( 2 \)[/tex].
### Conclusion
Given that we need to identify the function which has a growth factor of [tex]\( 5 \)[/tex]:
- The function with the growth factor of [tex]\( 5 \)[/tex] is [tex]\( f(x) = 2 \cdot 5^x \)[/tex].
Thus, the exponential function that has a growth factor of 5 is [tex]\( f(x) = 2 \cdot 5^x \)[/tex].
