What are the domain and range of [tex]$f(x)=\left(\frac{1}{5}\right)^x$[/tex]?

A. The domain is all real numbers. The range is all real numbers.

B. The domain is all real numbers. The range is all real numbers greater than zero.

C. The domain is all real numbers greater than zero. The range is all real numbers.

D. The domain is all real numbers greater than zero. The range is all real numbers greater than zero.



Answer :

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.