What is the domain of the given exponential function below? \( 2\left(\frac{1}{2}\right)^{n+1}+2 \) \( (2, \infty) \) \( [2, \infty) \) \( (-\infty, \infty) \) \( (1, \infty) \)

To determine the domain of the given exponential function \( f(n) = 2\left(\frac{1}{2}\right)^{n+1}+2 \), we need to analyze the function. 1. **Identify the function type**: The function is an exponential function, which is defined for all real numbers. The base \( \frac{1}{2} \) is a positive number, and the exponent \( n+1 \) can take any real value. 2. **Exponential function properties**: Exponential functions of the form \( a^x \) (where \( a > 0 \)) are defined for all real numbers \( x \). Therefore, \( \left(\frac{1}{2}\right)^{n+1} \) is defined for all \( n \). 3. **Adding constants**: The function also includes a constant term \( +2 \), which does not affect the domain. Since there are no restrictions on \( n \) in the expression, the domain of the function is all real numbers. Thus, the domain of the function \( 2\left(\frac{1}{2}\right)^{n+1}+2 \) is: \[ (-\infty, \infty) \] The correct answer is \( (-\infty, \infty) \).