What is the domain of $f(x)=5^x-7$?
A. {x | x>-7}
B. {x | x<-7}
C. {x | x>0}
D. {x | x is a real number }
Answer (1)
The function is f ( x ) = 5 x − 7 .
The exponential function 5 x is defined for all real numbers.
Subtracting a constant does not change the domain.
The domain of f ( x ) is all real numbers: { x ∣ x is a real number } .
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = 5 x − 7 . The domain of a function is the set of all possible values of x for which the function is defined.
Analyzing the Function The function f ( x ) = 5 x − 7 involves an exponential term 5 x and a constant term − 7 . The exponential function 5 x is defined for all real numbers x . Subtracting a constant from an exponential function does not change its domain.
Determining the Domain Therefore, the domain of f ( x ) = 5 x − 7 is all real numbers.
Final Answer The domain of f ( x ) = 5 x − 7 is the set of all real numbers, which can be written as { x ∣ x is a real number } .
Examples
Understanding the domain of functions is crucial in many real-world applications. For example, when modeling population growth with an exponential function, the domain represents the time frame over which the model is valid. Similarly, in finance, the domain of a function describing investment returns might be restricted by regulatory constraints or market conditions. Recognizing the domain helps ensure that the model's inputs are meaningful and the outputs are realistic.
