To determine which functions represent a stretch of an exponential growth function, we need to analyze the base of each function.
An exponential growth function has the form [tex]\( f(x) = a b^x \)[/tex] where the base [tex]\( b \)[/tex] is greater than 1.
Let's examine each of the given functions:
1. [tex]\( f(x) = \frac{2}{3}\left(\frac{2}{3}\right)^x \)[/tex]
- In this function, the base is [tex]\( \frac{2}{3} \)[/tex].
- Since [tex]\( \frac{2}{3} < 1 \)[/tex], this function represents exponential decay, not growth.
2. [tex]\( f(x) = \frac{3}{2}\left(\frac{2}{3}\right)^x \)[/tex]
- In this function, the base is [tex]\( \frac{2}{3} \)[/tex].
- Again, [tex]\( \frac{2}{3} < 1 \)[/tex], indicating that this function represents exponential decay.
3. [tex]\( f(x) = \frac{3}{2}\left(\frac{3}{2}\right)^x \)[/tex]
- Here, the base is [tex]\( \frac{3}{2} \)[/tex].
- Since [tex]\( \frac{3}{2} > 1 \)[/tex], this function represents exponential growth.
4. [tex]\( f(x) = \frac{2}{3}\left(\frac{3}{2}\right)^x \)[/tex]
- In this function, the base is [tex]\( \frac{3}{2} \)[/tex].
- As [tex]\( \frac{3}{2} > 1 \)[/tex], this function also represents exponential growth.
Therefore, the functions that represent exponential growth are:
[tex]\[ f(x) = \frac{3}{2}\left(\frac{3}{2}\right)^x \][/tex]
[tex]\[ f(x) = \frac{2}{3}\left(\frac{3}{2}\right)^x \][/tex]
Hence, the options that represent exponential growth functions are 3 and 4.
