To determine the domain of the exponential function [tex]\( f(x) = 2 \cdot \left( \frac{1}{10} \right)^x \)[/tex], we need to understand the properties of exponential functions.
The function [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b \)[/tex] is a positive real number, is an exponential function. For such functions, the base [tex]\( b \)[/tex] can be any positive number other than 1. In our given function, the base is [tex]\( \frac{1}{10} \)[/tex], which is a valid positive real number and not equal to 1.
The crucial feature of exponential functions is that they are defined for every real number [tex]\( x \)[/tex]. This means that, regardless of the exponent, the function will always produce a valid output real number.
Therefore, the variable [tex]\( x \)[/tex] in the function [tex]\( f(x) = 2 \cdot \left( \frac{1}{10} \right)^x \)[/tex] can take any real value. There are no restrictions on [tex]\( x \)[/tex] in the domain of the function.
So, the domain of the function [tex]\( f(x) = 2 \cdot \left( \frac{1}{10} \right)^x \)[/tex] is:
[tex]\[ \text{All real numbers} \][/tex]
Given the options:
A. [tex]\( x < 0 \)[/tex]
B. All real numbers
C. All real numbers except 2
D. [tex]\( x > 0 \)[/tex]
The correct answer is:
[tex]\[ \boxed{\text{B. All real numbers}} \][/tex]
