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If [tex]$f(x)=\left(\frac{1}{9}\right)\left(g^x\right)$[/tex], what is [tex]$f(3)$[/tex]?

A. [tex]$\frac{1}{729}$[/tex]

B. [tex]$\frac{1}{81}$[/tex]

C. 729

D. 81

Answer :

Let's solve this problem step-by-step!

Given the function:

[tex]\[ f(x) = \left(\frac{1}{9}\right)\left(g^x\right) \][/tex]

We need to find the value of [tex]\( f(3) \)[/tex].

1. Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[ f(3) = \frac{1}{9} \times g^3 \][/tex]

2. Since [tex]\( g \)[/tex] is not given a specific value in the problem, we typically either attempt to solve for [tex]\( g \)[/tex] using additional information or assume a plausible value for solving the problem with the given choices. Here, we assume [tex]\( g = 3 \)[/tex].

3. Update the expression with this assumption:
[tex]\[ f(3) = \frac{1}{9} \times 3^3 \][/tex]

4. Calculate [tex]\( 3^3 \)[/tex]:
[tex]\[ 3^3 = 27 \][/tex]

5. Now, substitute back to find [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = \frac{1}{9} \times 27 \][/tex]

6. Simplify the expression:
[tex]\[ f(3) = \frac{27}{9} = 3 \][/tex]

Therefore, the value of [tex]\( f(3) \)[/tex] is 3. None of the answer choices (A. [tex]\(\frac{1}{729}\)[/tex], B. [tex]\(\frac{1}{81}\)[/tex], C. 729, D. 81) matches this result directly, but with the assumption used, we solved according to the queried scenario.

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