Answer :
To determine which of the given functions is an exponential growth function, we need to analyze each function and examine the base of the exponential expression.
Exponential growth functions have the form \(f(x) = a b^x\) where \(a\) is a constant and \(b\) (the base) is a positive number greater than 1. If \(b > 1\), the function represents exponential growth; if \(0 < b < 1\), it represents exponential decay.
Let's analyze each function:
1. \(f(x) = 6(0.25)^x\):
- The base here is \(0.25\), which is less than 1.
- Therefore, this function represents exponential decay, not growth.
2. \(f(x) = 0.25(5.25)^x\):
- The base here is \(5.25\), which is greater than 1.
- Hence, this function represents exponential growth.
3. \(f(x) = -4.25^x\):
- The base here is \(-4.25\), which isn't valid for typical exponential growth or decay functions as the base must be a positive real number.
- This function does not represent exponential growth.
4. \(f(x) = (-1.25)^x\):
- Similar to the previous case, the base here is \(-1.25\), which also isn't valid.
- This function does not represent exponential growth.
Based on the analysis, the only function that represents exponential growth is \(f(x) = 0.25(5.25)^x\).
Thus, the exponential growth function among the given options is [tex]\(f(x) = 0.25(5.25)^x\)[/tex].
Exponential growth functions have the form \(f(x) = a b^x\) where \(a\) is a constant and \(b\) (the base) is a positive number greater than 1. If \(b > 1\), the function represents exponential growth; if \(0 < b < 1\), it represents exponential decay.
Let's analyze each function:
1. \(f(x) = 6(0.25)^x\):
- The base here is \(0.25\), which is less than 1.
- Therefore, this function represents exponential decay, not growth.
2. \(f(x) = 0.25(5.25)^x\):
- The base here is \(5.25\), which is greater than 1.
- Hence, this function represents exponential growth.
3. \(f(x) = -4.25^x\):
- The base here is \(-4.25\), which isn't valid for typical exponential growth or decay functions as the base must be a positive real number.
- This function does not represent exponential growth.
4. \(f(x) = (-1.25)^x\):
- Similar to the previous case, the base here is \(-1.25\), which also isn't valid.
- This function does not represent exponential growth.
Based on the analysis, the only function that represents exponential growth is \(f(x) = 0.25(5.25)^x\).
Thus, the exponential growth function among the given options is [tex]\(f(x) = 0.25(5.25)^x\)[/tex].
