To determine the domain of the function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex], we must identify the set of all possible values of [tex]\( x \)[/tex] for which the function is defined.
1. Analyzing the function form:
- The function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex] is an exponential function where the base is [tex]\( \frac{1}{2} \)[/tex].
- Exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a \)[/tex] is a positive constant) are defined for all real numbers [tex]\( x \)[/tex].
2. Understanding the base:
- The base [tex]\( \frac{1}{2} \)[/tex] is a positive number (since [tex]\( \frac{1}{2} > 0 \)[/tex]).
- There are no restrictions on the exponent [tex]\( x \)[/tex] in the function [tex]\( \left( \frac{1}{2} \right)^x \)[/tex].
3. Conclusion about the domain:
- Since [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] is defined for any real number [tex]\( x \)[/tex], the domain of the function is all real numbers.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{D. All real numbers}} \][/tex]
