To determine the range of the function [tex]\( F(x) = 6 + 2^x \)[/tex], let's analyze its behavior step by step.
1. Understanding the Function: The given function is [tex]\( F(x) = 6 + 2^x \)[/tex].
- Here, [tex]\( 6 \)[/tex] is a constant term.
- [tex]\( 2^x \)[/tex] is an exponential function where the base is 2.
2. Behavior of the Exponential Function:
- The exponential function [tex]\( 2^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex].
- Specifically, [tex]\( 2^x \)[/tex] ranges from [tex]\( 0 \)[/tex] (as [tex]\( x \to -\infty \)[/tex]) to [tex]\( +\infty \)[/tex].
3. Adding the Constant:
- Because [tex]\( 2^x \)[/tex] is always positive, [tex]\( 2^x > 0 \)[/tex].
- Adding 6 to [tex]\( 2^x \)[/tex] shifts the entire function upwards by 6 units.
- Therefore, [tex]\( 6 + 2^x \)[/tex] is always greater than 6.
4. Range Determination:
- Since [tex]\( 2^x \)[/tex] starts from a value just greater than 0 and goes to [tex]\( +\infty \)[/tex], adding 6 means [tex]\( 6 + 2^x \)[/tex] starts from a value just greater than 6 and also extends to [tex]\( +\infty \)[/tex].
- Thus, [tex]\( F(x) = 6 + 2^x \)[/tex] will take all values greater than 6 but never exactly equal to 6.
5. Conclusion:
- The range of the function [tex]\( F(x) = 6 + 2^x \)[/tex] is all real numbers greater than 6.
- Therefore, the given statement "The range of the function [tex]\( F(x) = 6 + 2^x \)[/tex] is the set of all positive real numbers greater than 6" is True.
Hence, the correct answer is:
A. True
