To determine whether the function [tex]\( F(x) = \log_{0.5}(x) \)[/tex] is increasing or not, we need to understand the behavior of logarithmic functions based on the base of the logarithm.
1. Logarithmic Function Basics:
- A logarithmic function [tex]\( \log_b(x) \)[/tex] has a base [tex]\( b \)[/tex].
- The function [tex]\( \log_b(x) \)[/tex] is increasing if [tex]\( b > 1 \)[/tex].
- The function [tex]\( \log_b(x) \)[/tex] is decreasing if [tex]\( 0 < b < 1 \)[/tex].
2. Given Problem:
- Here, the base [tex]\( b \)[/tex] is 0.5.
3. Behavior of Logarithmic Function:
- Since the base of the logarithm in [tex]\( F(x) = \log_{0.5}(x) \)[/tex] is 0.5, which lies between 0 and 1, the function is decreasing.
Thus, the statement "The function [tex]\( F(x) = \log_{0.5}(x) \)[/tex] is increasing" is false.
So, the correct answer is:
B. False
