To determine which of the given functions are examples of exponential growth, let's analyze each function individually.
### Function I:
[tex]\[ f(x) = \frac{1}{4} \left( \frac{1}{2} \right)^x \][/tex]
In this function, the base of the exponent is [tex]\( \frac{1}{2} \)[/tex].
- Exponential growth occurs when the base of the exponent is greater than 1.
- Here, [tex]\( \frac{1}{2} < 1 \)[/tex].
- A base less than 1 indicates exponential decay, not growth.
Hence, Function I is not an example of exponential growth.
### Function II:
[tex]\[ f(x) = 2 \left( 3^x \right) \][/tex]
In this function, the base of the exponent is [tex]\( 3 \)[/tex].
- Exponential growth occurs when the base of the exponent is greater than 1.
- Here, [tex]\( 3 > 1 \)[/tex], which means the function will grow exponentially as [tex]\( x \)[/tex] increases.
Hence, Function II is an example of exponential growth.
### Function III:
[tex]\[ f(x) = 5 \left( \frac{1}{2} \right)^x \][/tex]
In this function, the base of the exponent is [tex]\( \frac{1}{2} \)[/tex].
- Similar to Function I, a base of [tex]\( \frac{1}{2} \)[/tex] indicates exponential decay, not growth.
- Since [tex]\( \frac{1}{2} < 1 \)[/tex], the function decreases as [tex]\( x \)[/tex] increases.
Hence, Function III is not an example of exponential growth.
Based on our analysis, only Function II represents exponential growth. Therefore, the correct answer is:
B. II only
