Answer :
To determine which of the given functions represents exponential growth, we need to examine the base of the exponential part of each function. An exponential growth function has a base greater than 1.
Let’s analyze each function:
1. [tex]\( f(x) = 6 (0.25)^x \)[/tex]
- The base of the exponent is [tex]\( 0.25 \)[/tex].
- Since [tex]\( 0.25 \)[/tex] is less than 1, this is an exponential decay function, not a growth function.
2. [tex]\( f(x) = 0.25 (5.25)^x \)[/tex]
- The base of the exponent is [tex]\( 5.25 \)[/tex].
- Since [tex]\( 5.25 \)[/tex] is greater than 1, this is an exponential growth function.
3. [tex]\( f(x) = -4.25^x \)[/tex]
- The base of the exponent is [tex]\( -4.25 \)[/tex].
- Since the base is negative, this does not represent a traditional exponential growth or decay function.
4. [tex]\( f(x) = (-1.25)^x \)[/tex]
- The base of the exponent is [tex]\( -1.25 \)[/tex].
- Despite being negative, this function can behave unpredictably depending on whether [tex]\( x \)[/tex] is an integer or not, and generally it's not considered exponential growth.
Among all the functions, the only function that has a base greater than 1 (which makes it an exponential growth function) is:
[tex]\[ f(x) = 0.25 (5.25)^x \][/tex]
Therefore, the exponential growth function is:
[tex]\[ f(x) = 0.25 (5.25)^x \][/tex]
The corresponding answer from the given choices is the second function. Hence, the answer is:
[tex]\[ f(x) = 0.25 (5.25)^x \][/tex]
This corresponds to choice 3.
Let’s analyze each function:
1. [tex]\( f(x) = 6 (0.25)^x \)[/tex]
- The base of the exponent is [tex]\( 0.25 \)[/tex].
- Since [tex]\( 0.25 \)[/tex] is less than 1, this is an exponential decay function, not a growth function.
2. [tex]\( f(x) = 0.25 (5.25)^x \)[/tex]
- The base of the exponent is [tex]\( 5.25 \)[/tex].
- Since [tex]\( 5.25 \)[/tex] is greater than 1, this is an exponential growth function.
3. [tex]\( f(x) = -4.25^x \)[/tex]
- The base of the exponent is [tex]\( -4.25 \)[/tex].
- Since the base is negative, this does not represent a traditional exponential growth or decay function.
4. [tex]\( f(x) = (-1.25)^x \)[/tex]
- The base of the exponent is [tex]\( -1.25 \)[/tex].
- Despite being negative, this function can behave unpredictably depending on whether [tex]\( x \)[/tex] is an integer or not, and generally it's not considered exponential growth.
Among all the functions, the only function that has a base greater than 1 (which makes it an exponential growth function) is:
[tex]\[ f(x) = 0.25 (5.25)^x \][/tex]
Therefore, the exponential growth function is:
[tex]\[ f(x) = 0.25 (5.25)^x \][/tex]
The corresponding answer from the given choices is the second function. Hence, the answer is:
[tex]\[ f(x) = 0.25 (5.25)^x \][/tex]
This corresponds to choice 3.
