Answer :
To determine the correct point that lies on the graph of the function [tex]\( f(x) = \log_2 x \)[/tex], let's analyze the given information step by step.
1. Given Information:
- The point [tex]\( (-1, 0.5) \)[/tex] lies on the graph of the inverse function [tex]\( f^{-1}(x) = 2^x \)[/tex].
2. Understanding the Inverse:
- The inverse function [tex]\( f^{-1}(x) = 2^x \)[/tex] means the original function [tex]\( f(x) \)[/tex] is [tex]\( \log_2 x \)[/tex], as the logarithm base 2 is the inverse operation of the exponentiation with base 2.
- If a point [tex]\((a, b)\)[/tex] lies on the graph of the inverse function [tex]\( f^{-1} \)[/tex], then the point [tex]\((b, a)\)[/tex] lies on the graph of the original function [tex]\( f \)[/tex].
3. Reversing Roles of the Point:
- Given that the point [tex]\( (-1, 0.5) \)[/tex] lies on [tex]\( f^{-1}(x) = 2^x \)[/tex],
- Based on the properties of inverse functions, the point [tex]\( (0.5, -1) \)[/tex] will lie on the graph of the original function [tex]\( f(x) = \log_2 x \)[/tex].
4. Conclusion:
- The point that lies on [tex]\( f(x) = \log_2 x \)[/tex] is [tex]\( (0.5, -1) \)[/tex].
Therefore, the correct answer is
[tex]\[ \boxed{(0.5, -1)} \][/tex]
1. Given Information:
- The point [tex]\( (-1, 0.5) \)[/tex] lies on the graph of the inverse function [tex]\( f^{-1}(x) = 2^x \)[/tex].
2. Understanding the Inverse:
- The inverse function [tex]\( f^{-1}(x) = 2^x \)[/tex] means the original function [tex]\( f(x) \)[/tex] is [tex]\( \log_2 x \)[/tex], as the logarithm base 2 is the inverse operation of the exponentiation with base 2.
- If a point [tex]\((a, b)\)[/tex] lies on the graph of the inverse function [tex]\( f^{-1} \)[/tex], then the point [tex]\((b, a)\)[/tex] lies on the graph of the original function [tex]\( f \)[/tex].
3. Reversing Roles of the Point:
- Given that the point [tex]\( (-1, 0.5) \)[/tex] lies on [tex]\( f^{-1}(x) = 2^x \)[/tex],
- Based on the properties of inverse functions, the point [tex]\( (0.5, -1) \)[/tex] will lie on the graph of the original function [tex]\( f(x) = \log_2 x \)[/tex].
4. Conclusion:
- The point that lies on [tex]\( f(x) = \log_2 x \)[/tex] is [tex]\( (0.5, -1) \)[/tex].
Therefore, the correct answer is
[tex]\[ \boxed{(0.5, -1)} \][/tex]
