What is the domain of [tex]f(x)=\left(\frac{1}{4}\right)^x[/tex]?
A. All real numbers
B. [tex]x<0[/tex]
C. [tex]y>0[/tex]
D. [tex]x>0[/tex]
Answer (2)
The function is an exponential function of the form f ( x ) = a x .
Exponential functions are defined for all real numbers.
Therefore, the domain of f ( x ) = ( 4 1 ) x is all real numbers.
The answer is All real numbers .
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = ( 4 1 ) x . The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Domain of Exponential Functions The function f ( x ) = ( 4 1 ) x is an exponential function. Exponential functions are defined for all real numbers. This means that we can plug in any real number for x and get a real number output for f ( x ) .
Conclusion Therefore, the domain of f ( x ) = ( 4 1 ) x is all real numbers.
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 in the account after a certain period of time can be modeled by an exponential function. Understanding the domain of exponential functions helps us to determine the possible values for the independent variable (e.g., time) in these models.
The domain of the function f ( x ) = ( 4 1 ) x is all real numbers because exponential functions are defined for all real inputs. Thus, the correct answer is All real numbers .
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