Let's analyze the function [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex].
### Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex] is an exponential function. Exponential functions are defined for all real numbers, meaning you can input any real number for [tex]\( x \)[/tex] and the function will still produce a valid output.
So, the domain of [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex] is all real numbers.
### Range:
The range of a function is the set of all possible output values (y-values) the function can produce.
For [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex]:
- As [tex]\( x \to \infty \)[/tex] (x becomes very large), [tex]\( \left(\frac{1}{5}\right)^x \)[/tex] approaches 0 but never actually reaches 0. It remains positive no matter how large [tex]\( x \)[/tex] gets.
- As [tex]\( x \to -\infty \)[/tex] (x becomes very small), [tex]\( \left(\frac{1}{5}\right)^x \)[/tex] increases without bound because the base [tex]\( \frac{1}{5} \)[/tex] raised to a large negative power becomes a very large positive number.
Hence, the output values (y-values) of the function [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex] are always positive and can be any positive real number but never zero or negative.
So, the range of [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex] is all real numbers greater than zero.
### Conclusion:
- The domain is all real numbers.
- The range is all real numbers greater than zero.
Therefore, the correct answer is:
The domain is all real numbers. The range is all real numbers greater than zero.
