To determine the domain of the function [tex]\( f(x) = -\frac{5}{6} \left(\frac{3}{5}\right)^x \)[/tex], let's analyze it step by step.
1. Identify the type of function:
The function [tex]\( f(x) = -\frac{5}{6} \left(\frac{3}{5}\right)^x \)[/tex] is an exponential function. In general, exponential functions are written in the form [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.
2. Understand the components:
- The base of the exponent [tex]\( \left(\frac{3}{5}\right) \)[/tex] is a positive fraction.
- The coefficient [tex]\( -\frac{5}{6} \)[/tex] is a constant factor that scales and reflects the function but does not affect the domain.
3. Determine the domain of exponential functions:
Exponential functions of the form [tex]\( a \cdot b^x \)[/tex] are defined for all real numbers because you can raise a positive number, [tex]\( b \)[/tex], to any real power, [tex]\( x \)[/tex], without restriction.
4. Combine the observations:
Since [tex]\( \left(\frac{3}{5}\right)^x \)[/tex] is defined for all real numbers and the constant [tex]\( -\frac{5}{6} \)[/tex] does not impose any restrictions, the overall function [tex]\( f(x) = -\frac{5}{6} \left(\frac{3}{5}\right)^x \)[/tex] is also defined for all real numbers [tex]\( x \)[/tex].
Therefore, the domain of the function [tex]\( f(x) = -\frac{5}{6} \left(\frac{3}{5}\right)^x \)[/tex] is:
[tex]\[
\boxed{\text{All real numbers}}
\][/tex]
