Question
What is the domain of \( f(x)=5^{x}-7 \) ? \( \{x \mid x>-7\} \) \( \{x \mid x<-7\} \) \( \{x \mid x>0\} \) \( \{x \mid x \) is a real number \( \} \)
Ask by Horton Nunez. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of \( f(x) = 5^x - 7 \) is all real numbers.
**Simplified Answer:**
All real numbers
Solution
To find the domain of the function \( f(x) = 5^x - 7 \), we need to determine the values of \( x \) for which the function is defined.
1. The term \( 5^x \) is an exponential function. Exponential functions are defined for all real numbers. Therefore, \( 5^x \) can take any real number as input.
2. Since \( 5^x \) is defined for all real numbers, the expression \( 5^x - 7 \) is also defined for all real numbers.
Thus, the domain of \( f(x) = 5^x - 7 \) is:
\[
\{x \mid x \text{ is a real number}\}
\]
This means the correct answer is:
\(\{x \mid x \text{ is a real number} \}\)
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Simplify this solution Beyond the Answer
The domain of the function \( f(x) = 5^{x} - 7 \) is comprised of all real numbers. The exponential function \( 5^{x} \) is defined for every real number \( x \), meaning no restrictions are placed on the input values. Therefore, the correct choice here is \( \{x \mid x \text{ is a real number} \} \). Exponential functions like \( 5^x \) are commonly used in fields like finance for modeling growth or decay, such as compounded interest in investments. For instance, if you want to know how much an investment grows over time, you can use this function to predict its value!
