What is the domain of [tex]f(x)=5^x-7[/tex]?
A. [tex] \{x \mid x\ \textgreater \ -7\} [/tex]
B. [tex] \{x \mid x\ \textless \ -7\} [/tex]
C. [tex] \{x \mid x\ \textgreater \ 0\} [/tex]
D. [tex] \{x \mid x \text{ is a real number }\} [/tex]
Answer (2)
The function is f ( x ) = 5 x − 7 .
The exponential function 5 x is defined for all real numbers.
The constant term − 7 is defined for all real numbers.
The domain of f ( x ) is all real numbers: { x ∣ x is a real number } .
Explanation
Understanding the Function The function given is f ( x ) = 5 x − 7 . We need to determine the domain of this function, which is the set of all possible x values for which the function is defined.
Analyzing the Components The function f ( x ) consists of two parts: an exponential term 5 x and a constant term − 7 . Let's analyze each part separately.
Domain of the Exponential Term The exponential function 5 x is defined for all real numbers x . There are no restrictions on the values of x that can be used as an exponent.
Domain of the Constant Term The constant term − 7 is also defined for all real numbers. It does not impose any restrictions on the domain of the function.
Determining the Domain of f(x) Since both the exponential term and the constant term are defined for all real numbers, the function f ( x ) = 5 x − 7 is also defined for all real numbers. Therefore, the domain of f ( x ) is all real numbers.
Final Answer The domain of f ( x ) = 5 x − 7 is all real numbers, which can be written as { x ∣ x is a real number } .
Examples
Exponential functions are used in various real-world applications, such as modeling 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 after a certain period of time can be modeled using an exponential function. Understanding the domain of exponential functions helps you determine the possible values for the time variable in these models.
The domain of the function f ( x ) = 5 x − 7 is all real numbers, which can be expressed as { x ∣ x is a real number } . Therefore, the correct multiple-choice option is D. { x ∣ x is a real number } .
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