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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}{81}[/tex]
B. 729
C. [tex]\frac{1}{729}[/tex]
D. 81

Answer :

To find [tex]\( f(3) \)[/tex] for the function [tex]\( f(x) = \left(\frac{1}{9}\right) \left(g^x\right) \)[/tex], follow these steps:

1. Identify the Base [tex]\( g \)[/tex]: The function involves a base [tex]\( g \)[/tex] raised to the power of [tex]\( x \)[/tex]. According to the given problem, we realize that we should choose a value of [tex]\( g \)[/tex].

2. Calculate [tex]\( f(3) \)[/tex]: Since [tex]\( f(x) = \left(\frac{1}{9}\right) \left(g^x\right) \)[/tex], we substitute [tex]\( x = 3 \)[/tex]:

[tex]\[
f(3) = \left(\frac{1}{9}\right) \left(g^3\right)
\][/tex]

3. Determine [tex]\( g^3 \)[/tex] from Options:
- In the list of choices, if we consider [tex]\( g = 9 \)[/tex], then [tex]\( g^3 = 9^3 = 729 \)[/tex].

4. Substitute [tex]\( g^3 \)[/tex] into the Function Definition:
[tex]\[
f(3) = \left(\frac{1}{9}\right) \times 729
\][/tex]

5. Compute the Result:
- To find [tex]\( f(3) \)[/tex], calculate [tex]\( \frac{729}{9} \)[/tex].

[tex]\[
\frac{729}{9} = 81
\][/tex]

6. Conclude:
- Therefore, the value of [tex]\( f(3) \)[/tex] is [tex]\( 81 \)[/tex].

The correct answer is D. 81.

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