To determine the domain of the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex], we need to examine the properties of this exponential function.
1. Understanding the function:
The function [tex]\( f(x) \)[/tex] is of the form [tex]\( a^x \)[/tex], where [tex]\( a = \frac{1}{3} \)[/tex]. Exponential functions with a positive base other than 1 (i.e., [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) exhibit certain characteristics that are important for determining their domain.
2. Properties of exponential functions:
Exponential functions [tex]\( a^x \)[/tex] are defined for all real numbers [tex]\( x \)[/tex]. This means there is no restriction on [tex]\( x \)[/tex] that would cause the function to be undefined. For any real number [tex]\( x \)[/tex], we can calculate [tex]\( \left( \frac{1}{3} \right)^x \)[/tex].
3. Conclusion about the domain:
Since [tex]\( \left( \frac{1}{3} \right)^x \)[/tex] is defined for any real number [tex]\( x \)[/tex], the domain of [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex] is all real numbers.
Therefore, the correct option is:
C. All real numbers.
