To determine the domain and range of the function [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex], let's break down the problem step by step.
### Domain
The domain of a function refers to all possible input values (typically [tex]\( x \)[/tex] values) for which the function is defined. For the function [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex], note the following:
1. Exponential functions of the form [tex]\( a^x \)[/tex] where [tex]\( a > 0 \)[/tex] are defined for all real numbers [tex]\( x \)[/tex].
There are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( \left( \frac{1}{5} \right)^x \)[/tex]. This means that the domain of [tex]\( f(x) \)[/tex] is all real numbers.
So, the domain is: all real numbers.
### Range
The range of a function refers to all possible output values (typically [tex]\( y \)[/tex] values) that the function can take.
For [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex], we need to determine the behavior of the function as [tex]\( x \)[/tex] takes on various real numbers:
1. If [tex]\( x \)[/tex] is very large and positive, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] will approach 0, but it will never be exactly 0.
2. If [tex]\( x \)[/tex] is very large and negative, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] will grow very large, because raising [tex]\( \frac{1}{5} \)[/tex] to a large negative power is equivalent to raising 5 to a large positive power.
Thus, the function will yield positive values regardless of the input [tex]\( x \)[/tex], but it will never be zero or negative.
So the range is: all real numbers greater than zero.
### Conclusion
Combining our findings, the correct statements about the domain and range of [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex] are:
- 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.
