To determine the domain and range of the function [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex], we need to analyze the behavior of the function.
### Domain:
The domain refers to all the possible values that [tex]\( x \)[/tex] can take. In the function [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex], [tex]\( x \)[/tex] can be any real number.
- We can substitute any real number for [tex]\( x \)[/tex]. There are no restrictions such as division by zero or taking the square root of a negative number.
- Therefore, the domain is all real numbers.
### Range:
The range refers to all the possible values that [tex]\( f(x) \)[/tex] can take as [tex]\( x \)[/tex] varies over the domain.
- Observe the function [tex]\( f(x) = \left( \frac{1}{5} \right)^x \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = \left( \frac{1}{5} \right)^0 = 1 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] becomes very small but remains positive (i.e., [tex]\( 0 < \left( \frac{1}{5} \right)^x < 1 \)[/tex]).
- As [tex]\( x \)[/tex] decreases, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] becomes very large (i.e., [tex]\( x \to -\infty \implies \left( \frac{1}{5} \right)^x \to \infty \)[/tex]).
- No matter what real value [tex]\( x \)[/tex] takes, [tex]\( \left( \frac{1}{5} \right)^x \)[/tex] is always positive.
- Therefore, the function never reaches zero or any negative values.
- Thus, the range is all real numbers greater than zero.
Conclusively:
- The domain is all real numbers.
- The range is all real numbers greater than zero.
So, the correct answer is:
The domain is all real numbers. The range is all real numbers greater than zero.
