To determine which function is a stretch of an exponential growth function, we need to evaluate the coefficients and bases of the given exponential functions. A function represents a stretch of an exponential growth function if it has both a coefficient greater than 1 and a base greater than 1.
Let's analyze each function:
1. \( f(x) = \frac{2}{3}\left(\frac{2}{3}\right)^x \)
- Coefficient: \(\frac{2}{3}\)
- Base: \(\frac{2}{3}\)
- Both the coefficient and the base are less than 1. Hence, this function does not represent a stretch of an exponential growth function.
2. \( f(x) = \frac{3}{2}\left(\frac{2}{3}\right)^x \)
- Coefficient: \(\frac{3}{2}\)
- Base: \(\frac{2}{3}\)
- The coefficient is greater than 1, but the base is less than 1. Thus, this function does not represent a stretch of an exponential growth function.
3. \( f(x) = \frac{3}{2}\left(\frac{3}{2}\right)^x \)
- Coefficient: \(\frac{3}{2}\)
- Base: \(\frac{3}{2}\)
- Both the coefficient and the base are greater than 1. Therefore, this function does represent a stretch of an exponential growth function.
4. \( f(x) = \frac{2}{3}\left(\frac{3}{2}\right)^x \)
- Coefficient: \(\frac{2}{3}\)
- Base: \(\frac{3}{2}\)
- The base is greater than 1, but the coefficient is less than 1. Hence, this function does not represent a stretch of an exponential growth function.
Based on the analysis above, the function that represents a stretch of an exponential growth function is:
[tex]\[ f(x) = \frac{3}{2}\left(\frac{3}{2}\right)^x \][/tex]
Therefore, the correct option is the third one.
