The point $(-1,0.5)$ lies on the graph of $f^{-1}(x)=2^x$. Based on this information, which point lies on the graph of $f(x)=\log _2 x$?
A. $(-0.5,1)$
B. $(0.5,-1)$
C. $(1,-0.5)$
D. (1, 0.5)
Answer (1)
Given f − 1 ( x ) = 2 x and the point ( − 1 , 0.5 ) on its graph, we have f − 1 ( − 1 ) = 0.5 .
This implies that f ( 0.5 ) = − 1 .
Therefore, the point ( 0.5 , − 1 ) lies on the graph of f ( x ) = lo g 2 x .
The point on the graph of f ( x ) is ( 0.5 , − 1 ) .
Explanation
Understanding the Problem We are given that the point ( − 1 , 0.5 ) lies on the graph of f − 1 ( x ) = 2 x . This means that f − 1 ( − 1 ) = 0.5 . We want to find a point that lies on the graph of f ( x ) = lo g 2 x .
Using the Inverse Function Since f − 1 ( − 1 ) = 0.5 , this means that f ( 0.5 ) = − 1 . In other words, if we input 0.5 into the function f , we get − 1 as the output.
Finding the Point Therefore, the point ( 0.5 , − 1 ) lies on the graph of f ( x ) = lo g 2 x . We can verify this by checking if lo g 2 ( 0.5 ) = − 1 . Since 2 − 1 = 2 1 = 0.5 , this is true.
Final Answer The point that lies on the graph of f ( x ) = lo g 2 x is ( 0.5 , − 1 ) .
Examples
Understanding inverse functions is crucial in many real-world applications. For example, in cryptography, encoding and decoding messages rely on inverse functions. If f ( x ) encodes a message, then f − 1 ( x ) decodes it, ensuring secure communication. Similarly, in physics, if f ( x ) describes the position of an object at time x , then f − 1 ( x ) tells you the time at which the object was at position x . This concept is also used in computer graphics for transformations and inverse transformations.
