To evaluate the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex] for [tex]\( x = 3 \)[/tex], we will substitute 3 into the equation for [tex]\( x \)[/tex] and then simplify the expression.
Here are the steps:
1. Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex]:
[tex]\[
f(3) = \left( \frac{1}{3} \right)^3
\][/tex]
2. Simplify the exponentiation:
[tex]\[
\left( \frac{1}{3} \right)^3 = \frac{1^3}{3^3}
\][/tex]
3. Calculate the numerator and the denominator separately:
[tex]\[
1^3 = 1 \quad \text{and} \quad 3^3 = 27
\][/tex]
4. Place the results back into the fraction:
[tex]\[
\left( \frac{1}{3} \right)^3 = \frac{1}{27}
\][/tex]
Our final answer, as a reduced fraction, is:
[tex]\[
\boxed{\frac{1}{27}}
\][/tex]
