To determine whether the function [tex]\( f(x) = \frac{1}{2} \cdot 5^x \)[/tex] is an exponential function, we can analyze its structure.
1. Definition of an Exponential Function:
An exponential function can generally be written in the form [tex]\( f(x) = a \cdot b^x \)[/tex], where:
- [tex]\( a \)[/tex] is a constant.
- [tex]\( b \)[/tex] is a positive constant and [tex]\( b \neq 1 \)[/tex].
- [tex]\( x \)[/tex] is the exponent and the variable.
2. Identifying Constants in [tex]\( f(x) \)[/tex]:
Let's compare [tex]\( f(x) = \frac{1}{2} \cdot 5^x \)[/tex] with the exponential function format [tex]\( f(x) = a \cdot b^x \)[/tex]:
- Here, [tex]\( a = \frac{1}{2} \)[/tex].
- And [tex]\( b = 5 \)[/tex].
3. Checking the Form:
Since [tex]\( a = \frac{1}{2} \)[/tex] is a constant and [tex]\( b = 5 \)[/tex] is a positive constant different from 1, the given function [tex]\( f(x) = \frac{1}{2} \cdot 5^x \)[/tex] can indeed be written in the form [tex]\( f(x) = a \cdot b^x \)[/tex].
Therefore, based on the analysis, we can conclude:
- This is an exponential function because it can be written in the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b \)[/tex] is the base of the exponential function. Here, [tex]\( a = \frac{1}{2} \)[/tex] and [tex]\( b = 5 \)[/tex].
