To determine which of the given functions represents a stretch of an exponential growth function, let's analyze each function individually. An exponential growth function takes the form [tex]\( f(x) = ab^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential such that [tex]\( b > 1 \)[/tex].
1. Function: [tex]\( f(x) = \frac{2}{3}\left(\frac{2}{3}\right)^x \)[/tex]
- Here, [tex]\( a = \frac{2}{3} \)[/tex] and [tex]\( b = \frac{2}{3} \)[/tex].
- Since [tex]\( b = \frac{2}{3} \)[/tex] is less than 1, this function represents exponential decay, not exponential growth.
2. Function: [tex]\( f(x) = \frac{3}{2}\left(\frac{2}{3}\right)^x \)[/tex]
- Here, [tex]\( a = \frac{3}{2} \)[/tex] and [tex]\( b = \frac{2}{3} \)[/tex].
- Since [tex]\( b = \frac{2}{3} \)[/tex] is less than 1, this function also represents exponential decay.
3. Function: [tex]\( f(x) = \frac{3}{2}\left(\frac{3}{2}\right)^x \)[/tex]
- Here, [tex]\( a = \frac{3}{2} \)[/tex] and [tex]\( b = \frac{3}{2} \)[/tex].
- Since [tex]\( b = \frac{3}{2} \)[/tex] is greater than 1, this function represents exponential growth.
4. Function: [tex]\( f(x) = \frac{2}{3}\left(\frac{3}{2}\right)^x \)[/tex]
- Here, [tex]\( a = \frac{2}{3} \)[/tex] and [tex]\( b = \frac{3}{2} \)[/tex].
- While [tex]\( b = \frac{3}{2} \)[/tex] is greater than 1 and suggests exponential growth, the factor [tex]\( a = \frac{2}{3} \)[/tex] affects the initial value but does not complicate the nature of growth.
Out of these functions, the one that clearly represents a stretch of an exponential growth function, where both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are appropriately positioned for exponential growth, is:
[tex]\[ f(x) = \frac{3}{2}\left(\frac{3}{2}\right)^x \][/tex]
Therefore, the correct answer is option 3.
