Answer :
To determine the domain and range of the function [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex], let's analyze the properties of this exponential function step by step:
### Domain
The domain of a function refers to all the permissible values of [tex]\( x \)[/tex] that can be input into the function. For exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a \)[/tex] is a positive base), [tex]\( x \)[/tex] can be any real number. There are no restrictions, meaning you can raise [tex]\( \frac{1}{5} \)[/tex] to any power (positive, negative, or zero) and still obtain a valid real number.
Therefore, the domain of [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex] is all real numbers.
### Range
The range of a function refers to all the possible output values ([tex]\( y \)[/tex]-values) that the function can produce. For exponential functions with a positive base [tex]\( 0 < a < 1 \)[/tex], as [tex]\( x \)[/tex] increases, the value of [tex]\( a^x \)[/tex] decreases towards zero but never actually reaches zero. As [tex]\( x \)[/tex] decreases, the value of [tex]\( a^x \)[/tex] increases without bound.
Thus, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] will always be a positive number regardless of the value of [tex]\( x \)[/tex]. It can get arbitrarily close to zero but never actually be zero. Conversely, it can get very large but will never be negative.
Therefore, the range of [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex] is all real numbers greater than zero.
### Conclusion
Combining both the domain and range:
- The domain is all real numbers.
- The range is all real numbers greater than zero.
Hence, the correct answer is:
- The domain is all real numbers. The range is all real numbers greater than zero.
### Domain
The domain of a function refers to all the permissible values of [tex]\( x \)[/tex] that can be input into the function. For exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a \)[/tex] is a positive base), [tex]\( x \)[/tex] can be any real number. There are no restrictions, meaning you can raise [tex]\( \frac{1}{5} \)[/tex] to any power (positive, negative, or zero) and still obtain a valid real number.
Therefore, the domain of [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex] is all real numbers.
### Range
The range of a function refers to all the possible output values ([tex]\( y \)[/tex]-values) that the function can produce. For exponential functions with a positive base [tex]\( 0 < a < 1 \)[/tex], as [tex]\( x \)[/tex] increases, the value of [tex]\( a^x \)[/tex] decreases towards zero but never actually reaches zero. As [tex]\( x \)[/tex] decreases, the value of [tex]\( a^x \)[/tex] increases without bound.
Thus, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] will always be a positive number regardless of the value of [tex]\( x \)[/tex]. It can get arbitrarily close to zero but never actually be zero. Conversely, it can get very large but will never be negative.
Therefore, the range of [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex] is all real numbers greater than zero.
### Conclusion
Combining both the domain and range:
- The domain is all real numbers.
- The range is all real numbers greater than zero.
Hence, the correct answer is:
- The domain is all real numbers. The range is all real numbers greater than zero.
