To determine the domain of the function [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex], let's analyze the properties of exponential functions.
1. Identify the form of the function:
The function [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex] is an exponential function. Exponential functions generally have the form [tex]\( f(x) = a b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the exponent.
2. Understanding exponential functions:
Exponential functions are defined for all real numbers. This is because you can exponentiate a constant [tex]\( b \)[/tex] with any real number [tex]\( x \)[/tex] without any restrictions. The base [tex]\( \left( \frac{3}{5} \right) \)[/tex] is a positive constant and can be raised to any real power [tex]\( x \)[/tex].
3. Determine the domain:
Since exponential functions are defined for every real number [tex]\( x \)[/tex], the domain of the function [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex] is all real numbers. There are no restrictions on the values [tex]\( x \)[/tex] can take.
Thus, the domain of [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex] is:
[tex]\[
\boxed{\text{all real numbers}}
\][/tex]
