Answer :
To determine which of the given functions represents an exponential growth function, we need to analyze the base [tex]\( b \)[/tex] of the exponential term [tex]\( b^x \)[/tex]. An exponential growth function is characterized by a base [tex]\( b \)[/tex] where [tex]\( b > 1 \)[/tex].
Let's examine each function step by step:
1. [tex]\( f(x) = 6(0.25)^x \)[/tex]:
- The base of the exponential term here is [tex]\( 0.25 \)[/tex].
- Since [tex]\( 0.25 < 1 \)[/tex], this function represents exponential decay, not growth.
2. [tex]\( f(x) = 0.25(5.25)^x \)[/tex]:
- The base of the exponential term here is [tex]\( 5.25 \)[/tex].
- Since [tex]\( 5.25 > 1 \)[/tex], this function represents exponential growth.
3. [tex]\( f(x) = -4.25^x \)[/tex]:
- The base of the exponential term here is [tex]\( -4.25 \)[/tex].
- Generally, negative bases for exponents are not considered typical exponential growth functions, as the function can have complex and alternating behavior depending on whether [tex]\( x \)[/tex] is an integer or not.
4. [tex]\( f(x) = (-1.25)^x \)[/tex]:
- The base of the exponential term here is [tex]\( -1.25 \)[/tex].
- Similar to the previous case, negative bases are not treated as generic exponential growth functions due to potential complex and alternating behavior.
Among the given options, the function [tex]\( f(x) = 0.25(5.25)^x \)[/tex] is the one that represents exponential growth because the base [tex]\( 5.25 \)[/tex] is greater than [tex]\( 1 \)[/tex].
Therefore, the correct answer is:
Option 2: [tex]\( f(x) = 0.25(5.25)^x \)[/tex]
Let's examine each function step by step:
1. [tex]\( f(x) = 6(0.25)^x \)[/tex]:
- The base of the exponential term here is [tex]\( 0.25 \)[/tex].
- Since [tex]\( 0.25 < 1 \)[/tex], this function represents exponential decay, not growth.
2. [tex]\( f(x) = 0.25(5.25)^x \)[/tex]:
- The base of the exponential term here is [tex]\( 5.25 \)[/tex].
- Since [tex]\( 5.25 > 1 \)[/tex], this function represents exponential growth.
3. [tex]\( f(x) = -4.25^x \)[/tex]:
- The base of the exponential term here is [tex]\( -4.25 \)[/tex].
- Generally, negative bases for exponents are not considered typical exponential growth functions, as the function can have complex and alternating behavior depending on whether [tex]\( x \)[/tex] is an integer or not.
4. [tex]\( f(x) = (-1.25)^x \)[/tex]:
- The base of the exponential term here is [tex]\( -1.25 \)[/tex].
- Similar to the previous case, negative bases are not treated as generic exponential growth functions due to potential complex and alternating behavior.
Among the given options, the function [tex]\( f(x) = 0.25(5.25)^x \)[/tex] is the one that represents exponential growth because the base [tex]\( 5.25 \)[/tex] is greater than [tex]\( 1 \)[/tex].
Therefore, the correct answer is:
Option 2: [tex]\( f(x) = 0.25(5.25)^x \)[/tex]
