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 (1)
The domain of the exponential function f ( x ) = ( 5 1 ) x is all real numbers.
The range of the exponential function f ( x ) = ( 5 1 ) x is all real numbers greater than zero.
Therefore, the domain is all real numbers and the range is all real numbers greater than zero.
The final answer is: The domain is all real numbers. The range is all real numbers greater than zero. The domain is all real numbers. The range is all real numbers greater than zero.
Explanation
Understanding the Function We are asked to find the domain and range of the function f ( x ) = ( 5 1 ) x . This is an exponential function with base 5 1 .
Determining the Domain The domain of an exponential function a x , where a is a positive real number, is all real numbers. In this case, a = 5 1 , which is a positive real number. Therefore, the domain of f ( x ) = ( 5 1 ) x is all real numbers.
Determining the Range The range of an exponential function a x , where a is a positive real number not equal to 1, is all positive real numbers. As x approaches positive infinity, f ( x ) approaches 0, but never actually reaches 0. As x approaches negative infinity, f ( x ) approaches infinity. Therefore, the range of f ( x ) = ( 5 1 ) x is all real numbers greater than zero.
Final Answer Therefore, the domain of f ( x ) = ( 5 1 ) x is all real numbers, and the range is all real numbers greater than zero.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have in the account after a certain period of time can be modeled by an exponential function. Understanding the domain and range of exponential functions can help you make informed decisions about investments and other financial matters.
