To determine the domain of the function [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex], we need to consider the different parts of the function and the implications they have on the values [tex]\( x \)[/tex] can take.
The function [tex]\( f(x) \)[/tex] involves:
- A constant multiplier, [tex]\(-\frac{5}{6}\)[/tex].
- An exponential component where the base is [tex]\(\frac{3}{5}\)[/tex] and the exponent is [tex]\( x \)[/tex].
Let's carefully examine each part:
1. Constant Multiplier [tex]\(-\frac{5}{6}\)[/tex]:
This is just a constant factor and does not affect the domain of the function. A constant multiplier can be applied to any value without restriction.
2. Exponential Component [tex]\(\left( \frac{3}{5} \right)^x \)[/tex]:
- The base of the exponent [tex]\(\left( \frac{3}{5} \right)\)[/tex] is a positive rational number less than 1 ([tex]\(0 < \frac{3}{5} < 1\)[/tex]).
- The exponent [tex]\( x \)[/tex] can be any real number.
The exponential function [tex]\(\left( \frac{3}{5} \right)^x\)[/tex] is defined for all real numbers since raising a positive number to any real power is well-defined.
Therefore, the combination of a constant multiplier with a well-defined exponential function means the overall function [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex] is defined for all real numbers.
Conclusion:
The domain of the function [tex]\( f(x) = -\frac{5}{6} \left( \frac{3}{5} \right)^x \)[/tex] is all real numbers.
Thus, the answer is:
all real numbers
This ensures that [tex]\( x \)[/tex] can be any real number, and there are no restrictions on [tex]\( x \)[/tex] for the given function.
