Answer :
To identify the correct equivalent form of the function [tex]\( f \)[/tex] that displays [tex]\(\frac{1}{c}\)[/tex] as a coefficient of [tex]\( x \)[/tex], we need to analyze the given options.
The problem states that for each increase in [tex]\( x \)[/tex] by [tex]\( c \)[/tex], the value of [tex]\( f(x) \)[/tex] increases by a factor of 27. This implies that the function's growth is tied to multiplying by 27 when [tex]\( x \)[/tex] increases by [tex]\( c \)[/tex].
Let's examine each form and identify the value of [tex]\( c \)[/tex].
Option (A): [tex]\( f(x) = 48(3)^{\frac{1}{2} x} \)[/tex]
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{2}\)[/tex].
- If [tex]\( x \)[/tex] increases by [tex]\( 2 \)[/tex]:
[tex]\[ f(x + 2) = 48 \cdot (3)^{\frac{1}{2}(x + 2)} = 48 \cdot (3)^{\frac{1}{2}x + 1} = 48 \cdot (3)^{\frac{1}{2}x} \cdot 3 = 3f(x) \][/tex]
Since we require the function to increase by a factor of 27, not 3, this form is incorrect.
Option (B): [tex]\( f(x) = 48\left(3^3\right)^{\frac{1}{6} x} \)[/tex]
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{6}\)[/tex].
- If [tex]\( x \)[/tex] increases by [tex]\( 6 \)[/tex]:
[tex]\[ f(x + 6) = 48 \cdot \left(3^3\right)^{\frac{1}{6}(x + 6)} = 48 \cdot \left(3^3\right)^{\frac{1}{6}x + 1} = 48 \cdot \left(3^3\right)^{\frac{1}{6}x} \cdot 3 = 27f(x) \][/tex]
This matches the required factor of 27, indicating [tex]\( c = 6 \)[/tex]. Hence, [tex]\(\frac{1}{c} = \frac{1}{6}\)[/tex], which is displayed correctly in the function. Therefore, this form is correct.
Option (C): [tex]\( f(x) = 48(9)^{\frac{1}{4} x} \)[/tex]
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{4}\)[/tex].
- If [tex]\( x \)[/tex] increases by [tex]\( 4 \)[/tex]:
[tex]\[ f(x + 4) = 48 \cdot (9)^{\frac{1}{4}(x + 4)} = 48 \cdot (9)^{\frac{1}{4}x + 1} = 48 \cdot (9)^{\frac{1}{4}x} \cdot 9 = 9f(x) \][/tex]
Since we need the function to increase by a factor of 27, not 9, this form is incorrect.
Option (D): [tex]\( f(x) = 48\left(27^{\frac{1}{3} x}\right)^{\frac{1}{2}} \)[/tex]
- This can be simplified to [tex]\( f(x) = 48(27^{\frac{1}{6} x}) \)[/tex].
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{6}\)[/tex], similar to Option (B).
- If [tex]\( x \)[/tex] increases by [tex]\( 6 \)[/tex]:
[tex]\[ f(x + 6) = 48 \cdot (27)^{\frac{1}{6}(x + 6)} = 48 \cdot (27)^{\frac{1}{6}x + 1} = 48 \cdot (27)^{\frac{1}{6}x} \cdot 27 = 27f(x) \][/tex]
This also matches the required factor of 27, indicating [tex]\( c = 6 \)[/tex]. Hence, [tex]\(\frac{1}{c} = \frac{1}{6}\)[/tex], which is displayed correctly in the function. Therefore, this form is also correct.
Among the options, both Option (B) and Option (D) yield [tex]\(\frac{1}{c} = \frac{1}{6}\)[/tex]. However, since we only need one correct answer:
The option that displays [tex]\(\frac{1}{c}\)[/tex] as a coefficient of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{B} \][/tex]
The problem states that for each increase in [tex]\( x \)[/tex] by [tex]\( c \)[/tex], the value of [tex]\( f(x) \)[/tex] increases by a factor of 27. This implies that the function's growth is tied to multiplying by 27 when [tex]\( x \)[/tex] increases by [tex]\( c \)[/tex].
Let's examine each form and identify the value of [tex]\( c \)[/tex].
Option (A): [tex]\( f(x) = 48(3)^{\frac{1}{2} x} \)[/tex]
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{2}\)[/tex].
- If [tex]\( x \)[/tex] increases by [tex]\( 2 \)[/tex]:
[tex]\[ f(x + 2) = 48 \cdot (3)^{\frac{1}{2}(x + 2)} = 48 \cdot (3)^{\frac{1}{2}x + 1} = 48 \cdot (3)^{\frac{1}{2}x} \cdot 3 = 3f(x) \][/tex]
Since we require the function to increase by a factor of 27, not 3, this form is incorrect.
Option (B): [tex]\( f(x) = 48\left(3^3\right)^{\frac{1}{6} x} \)[/tex]
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{6}\)[/tex].
- If [tex]\( x \)[/tex] increases by [tex]\( 6 \)[/tex]:
[tex]\[ f(x + 6) = 48 \cdot \left(3^3\right)^{\frac{1}{6}(x + 6)} = 48 \cdot \left(3^3\right)^{\frac{1}{6}x + 1} = 48 \cdot \left(3^3\right)^{\frac{1}{6}x} \cdot 3 = 27f(x) \][/tex]
This matches the required factor of 27, indicating [tex]\( c = 6 \)[/tex]. Hence, [tex]\(\frac{1}{c} = \frac{1}{6}\)[/tex], which is displayed correctly in the function. Therefore, this form is correct.
Option (C): [tex]\( f(x) = 48(9)^{\frac{1}{4} x} \)[/tex]
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{4}\)[/tex].
- If [tex]\( x \)[/tex] increases by [tex]\( 4 \)[/tex]:
[tex]\[ f(x + 4) = 48 \cdot (9)^{\frac{1}{4}(x + 4)} = 48 \cdot (9)^{\frac{1}{4}x + 1} = 48 \cdot (9)^{\frac{1}{4}x} \cdot 9 = 9f(x) \][/tex]
Since we need the function to increase by a factor of 27, not 9, this form is incorrect.
Option (D): [tex]\( f(x) = 48\left(27^{\frac{1}{3} x}\right)^{\frac{1}{2}} \)[/tex]
- This can be simplified to [tex]\( f(x) = 48(27^{\frac{1}{6} x}) \)[/tex].
- Here, the coefficient of [tex]\( x \)[/tex] in the exponent is [tex]\(\frac{1}{6}\)[/tex], similar to Option (B).
- If [tex]\( x \)[/tex] increases by [tex]\( 6 \)[/tex]:
[tex]\[ f(x + 6) = 48 \cdot (27)^{\frac{1}{6}(x + 6)} = 48 \cdot (27)^{\frac{1}{6}x + 1} = 48 \cdot (27)^{\frac{1}{6}x} \cdot 27 = 27f(x) \][/tex]
This also matches the required factor of 27, indicating [tex]\( c = 6 \)[/tex]. Hence, [tex]\(\frac{1}{c} = \frac{1}{6}\)[/tex], which is displayed correctly in the function. Therefore, this form is also correct.
Among the options, both Option (B) and Option (D) yield [tex]\(\frac{1}{c} = \frac{1}{6}\)[/tex]. However, since we only need one correct answer:
The option that displays [tex]\(\frac{1}{c}\)[/tex] as a coefficient of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{B} \][/tex]
