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 (2)
The domain of the function f ( x ) = ( 5 1 ) x is all real numbers.
The range of the function f ( x ) = ( 5 1 ) x is all real numbers greater than zero.
Therefore, the domain is ( − ∞ , ∞ ) and the range is ( 0 , ∞ ) .
The correct answer is: The domain is all real numbers. The range is all real numbers greater than zero. $\boxed{The domain is all real numbers. The range is all real numbers greater than zero.}
Explanation
Understanding Domain and Range We are asked to find the domain and range of the exponential function f ( x ) = ( 5 1 ) x . Let's analyze what domain and range mean and how they apply to this function.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's what you're allowed to plug into the function. For exponential functions like this one, we can plug in any real number for x . 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.
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. For f ( x ) = ( 5 1 ) x , as x varies over all real numbers, the function will always produce a positive value. No matter how large or small x is, ( 5 1 ) x will never be zero or negative. As x approaches positive infinity, f ( x ) approaches 0, but never actually reaches it. As x approaches negative infinity, f ( x ) becomes infinitely large. Therefore, the range is all real numbers greater than zero.
Final Answer In summary, 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 a bank account that earns interest compounded annually, the amount of money you have after x years can be modeled by an exponential function. Understanding the domain and range of exponential functions helps you determine the possible values for the time (x) and the amount of money (y) in the account.
The function f ( x ) = ( 5 1 ) x has a domain of all real numbers, meaning you can input any real number for x . Its range is all positive real numbers, as the output will never be zero or negative. Therefore, the choice is B: The domain is all real numbers; the range is all real numbers greater than zero.
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