To determine the type of function represented by [tex]\( f(x) = 3(1.5)^x \)[/tex], we will analyze the properties of the function.
1. Identify the function type:
The given function is [tex]\( f(x) = 3(1.5)^x \)[/tex]. Notice that the function takes the form [tex]\( a(b)^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable. Functions of this form are typically exponential functions.
2. Exponential Functions:
Exponential functions can be recognized by their form [tex]\( f(x) = a \cdot b^x \)[/tex]:
- [tex]\( a \)[/tex] is a constant multiplier.
- [tex]\( b \)[/tex] is the base of the exponential.
- [tex]\( x \)[/tex] is the exponent.
3. Analyzing the base:
To determine whether the exponential function represents growth or decay, we need to look at the base [tex]\( b \)[/tex]:
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
In [tex]\( f(x) = 3(1.5)^x \)[/tex], the base [tex]\( b \)[/tex] is 1.5.
4. Determine growth or decay:
Since [tex]\( b = 1.5 \)[/tex] and [tex]\( 1.5 > 1 \)[/tex], this means the base is greater than 1. Therefore, the function is experiencing exponential growth.
5. Conclusion:
Based on the identified properties, we can conclude:
- The function [tex]\( f(x) = 3(1.5)^x \)[/tex] is an exponential function.
- Since the base 1.5 is greater than 1, it represents exponential growth.
Thus, the correct answer is:
A. Exponential growth
