To determine the domain of the function [tex]\( y = \left( \frac{1}{3} \right)^x \)[/tex], we need to consider the values of [tex]\( x \)[/tex] that are permissible in the function.
1. Understand the Function: The function [tex]\( y = \left( \frac{1}{3} \right)^x \)[/tex]:
- This is an exponential function, specifically in the form [tex]\( a^x \)[/tex] where [tex]\( a = \frac{1}{3} \)[/tex].
2. Domain of Exponential Functions: For exponential functions of the form [tex]\( f(x) = a^x \)[/tex], where [tex]\( a \)[/tex] is a positive constant, the exponent [tex]\( x \)[/tex] can be any real number. This is because there are no restrictions or limitations imposed by the base [tex]\( \frac{1}{3} \)[/tex].
3. Set of Possible x-values: Since [tex]\( x \)[/tex] can take any real number value without causing any mathematical inconsistencies (like division by zero, negative inside a logarithm, or taking the even root of a negative number), [tex]\( x \)[/tex] includes all real numbers.
Thus, the domain of the function [tex]\( y = \left( \frac{1}{3} \right)^x \)[/tex] is all real numbers.
We symbolize the set of all real numbers by [tex]\( R \)[/tex], so the domain is:
[tex]\[ \boxed{R} \][/tex]
