To determine the domain and range of the function [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex], let's analyze the characteristics of exponential functions, particularly those of the form [tex]\( a^x \)[/tex] where [tex]\( 0 < a < 1 \)[/tex].
### Domain
The domain of a function refers to all possible input values (x-values) that the function can accept. In general, exponential functions [tex]\( a^x \)[/tex] are defined for all real numbers [tex]\( x \)[/tex]. This means that there are no restrictions on the values that [tex]\( x \)[/tex] can take.
So, for [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex], the domain is:
All real numbers.
### Range
The range of a function refers to all possible output values (y-values) that the function can produce. Exponential functions of the form [tex]\( a^x \)[/tex] where [tex]\( 0 < a < 1 \)[/tex] have certain properties:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]), [tex]\( \left(\frac{1}{5}\right)^x \)[/tex] gets closer and closer to zero but never actually reaches zero.
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( \left(\frac{1}{5}\right)^x \)[/tex] grows larger and larger without bound.
Since the function never actually reaches zero and always produces a positive value for any real [tex]\( x \)[/tex], the range is:
All real numbers greater than zero.
### Conclusion
Based on the characteristics of the exponential function [tex]\( f(x) = \left(\frac{1}{5}\right)^x \)[/tex], the correct statement regarding the domain and range is:
- The domain is all real numbers.
- The range is all real numbers greater than zero.
Thus, the correct choice is:
The domain is all real numbers. The range is all real numbers greater than zero.
