College

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

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

Answer :

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

1. Understand the Function: The given function is [tex]\( f(x) = \left(\frac{1}{9}\right)\left(9^x\right) \)[/tex]. It involves a power function [tex]\( 9^x \)[/tex] which is then multiplied by [tex]\(\frac{1}{9}\)[/tex].

2. Substitute the Value: We need to find [tex]\( f(3) \)[/tex]. So, substitute [tex]\( x = 3 \)[/tex] into the function:

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

3. Calculate [tex]\( 9^3 \)[/tex]: First, calculate [tex]\( 9^3 \)[/tex].

[tex]\[
9^3 = 9 \times 9 \times 9 = 729
\][/tex]

4. Multiply by [tex]\(\frac{1}{9}\)[/tex]: Next, multiply 729 by [tex]\(\frac{1}{9}\)[/tex]:

[tex]\[
f(3) = \left(\frac{1}{9}\right)\times 729 = \frac{729}{9}
\][/tex]

5. Simplify the Expression: Divide 729 by 9:

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

Therefore, the value of [tex]\( f(3) \)[/tex] is [tex]\( 81 \)[/tex], which corresponds to option D.