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 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 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, that would limit the possible x -values.
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. For the function f ( x ) = ( 5 1 ) x , we need to consider what happens as x varies. As x gets very large (positive), ( 5 1 ) x gets very close to 0, but it never actually reaches 0. As x gets very small (negative), ( 5 1 ) x becomes very large (positive). Since the base 5 1 is a positive number, the function will always produce positive values. Therefore, the range is all positive real numbers.
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 compound interest, the amount of money you have over time can be modeled by an exponential function. Understanding the domain and range of these functions helps you make predictions about the future value of your investment.
The domain of the function f ( x ) = ( 5 1 ) x is all real numbers, and the range is all positive real numbers greater than zero. The correct answer is option B.
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