Certainly! Let's analyze the function [tex]\( f(x) = 4 \cdot 3^x \)[/tex] to determine its type.
1. General Form of Functions:
- Exponential functions generally have the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants. For exponential growth, [tex]\( b > 1 \)[/tex], and for exponential decay, [tex]\( 0 < b < 1 \)[/tex].
- Linear functions have the form [tex]\( f(x) = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
2. Given Function:
- The function provided is [tex]\( f(x) = 4 \cdot 3^x \)[/tex].
3. Analyzing the Form:
- Compare [tex]\( f(x) = 4 \cdot 3^x \)[/tex] to the general form of exponential functions, [tex]\( f(x) = a \cdot b^x \)[/tex].
- Here, [tex]\( a = 4 \)[/tex] and [tex]\( b = 3 \)[/tex].
4. Conditions for Exponential Growth:
- For the function to represent exponential growth, [tex]\( b \)[/tex] must be greater than 1.
- In our function, [tex]\( b = 3 \)[/tex], which is greater than 1.
Therefore, the function [tex]\( f(x) = 4 \cdot 3^x \)[/tex] fits the criteria for an exponential growth function because it is of the form [tex]\( a \cdot b^x \)[/tex] with [tex]\( b > 1 \)[/tex].
Conclusion:
The correct answer is:
A. Exponential growth
