If $f(x)$ and its inverse function, $f^{-1}(x)$, are both plotted on the same coordinate plane, what is their point of intersection?
$(0,-2)$
$(1,-1)$
Answer (2)
The intersection point of a function and its inverse lies on the line y = x .
Check if the given points satisfy the condition y = x .
The point ( 0 , − 2 ) does not satisfy y = x since 0 = − 2 , and the point ( 1 , − 1 ) does not satisfy y = x since 1 = − 1 .
Therefore, there must be an error in the problem statement or the given options. Neither of the points is a valid intersection point. There is no correct answer among the options. If forced to pick one, it would be neither. However, since the intersection must lie on y = x , the problem is flawed.
Explanation
Understanding the Problem Let's analyze the problem. We are given that f ( x ) and its inverse f − 1 ( x ) are plotted on the same coordinate plane, and we need to find their point of intersection. We are given two options: ( 0 , − 2 ) and ( 1 , − 1 ) .
Properties of Inverse Functions A key property of inverse functions is that if ( a , b ) is a point on the graph of f ( x ) , then ( b , a ) is a point on the graph of f − 1 ( x ) . If f ( x ) and f − 1 ( x ) intersect at a point ( x 0 , y 0 ) , then f ( x 0 ) = y 0 and f − 1 ( x 0 ) = y 0 . This implies that f ( y 0 ) = x 0 .
Intersection Points and the Line y=x If the intersection point lies on the line y = x , then x 0 = y 0 , and f ( x 0 ) = x 0 . In other words, the function value at x 0 is equal to x 0 . However, it is also possible for the graphs of f ( x ) and f − 1 ( x ) to intersect at a point that does not lie on the line y = x . In this case, if ( a , b ) is a point of intersection, then f ( a ) = b and f − 1 ( a ) = b , which means f ( b ) = a . So both ( a , b ) and ( b , a ) are points on f ( x ) .
Checking the Given Points Let's check if the given points satisfy the condition y = x . For the point ( 0 , − 2 ) , we have 0 = − 2 , so it does not lie on the line y = x . For the point ( 1 , − 1 ) , we have 1 = − 1 , so it does not lie on the line y = x . However, this doesn't mean these points cannot be intersection points. If ( 0 , − 2 ) is on f ( x ) , then ( − 2 , 0 ) must be on f − 1 ( x ) . If ( 1 , − 1 ) is on f ( x ) , then ( − 1 , 1 ) must be on f − 1 ( x ) .
Analyzing Possible Intersection Points If ( 0 , − 2 ) is the intersection point, then f ( 0 ) = − 2 and f − 1 ( 0 ) = − 2 . This means f ( − 2 ) = 0 . So, we have the points ( 0 , − 2 ) and ( − 2 , 0 ) . If ( 1 , − 1 ) is the intersection point, then f ( 1 ) = − 1 and f − 1 ( 1 ) = − 1 . This means f ( − 1 ) = 1 . So, we have the points ( 1 , − 1 ) and ( − 1 , 1 ) .
Considering Points Not on y=x Since neither of the given points lies on the line y = x , we need to consider the possibility that the intersection point ( a , b ) does not satisfy a = b . In this case, we must have f ( a ) = b and f ( b ) = a . If ( 0 , − 2 ) is the intersection, then f ( 0 ) = − 2 and f − 1 ( 0 ) = − 2 , so f ( − 2 ) = 0 . If ( 1 , − 1 ) is the intersection, then f ( 1 ) = − 1 and f − 1 ( 1 ) = − 1 , so f ( − 1 ) = 1 .
Final Conclusion Without additional information about the function f ( x ) , we cannot definitively determine the point of intersection. However, if we assume that the intersection point must lie on the line y = x , then neither of the given options is correct. If we don't make this assumption, both options are possible. However, the problem implies that there is a definite answer. The intersection of a function and its inverse must lie on the line y = x . Since neither point satisfies this, there must be an error in the problem statement.
Addressing the Options Since the intersection point must lie on the line y = x , we look for a point ( a , a ) such that f ( a ) = a . Neither ( 0 , − 2 ) nor ( 1 , − 1 ) satisfy this condition. Therefore, there must be an error in the problem statement or the given options. However, if we are forced to choose one of the given options, we can say that neither point is necessarily the intersection of f ( x ) and f − 1 ( x ) .
Examples
Understanding the intersection of a function and its inverse is crucial in many areas of mathematics and its applications. For example, in cryptography, the inverse function is used to decrypt messages that were encrypted using the original function. The point of intersection can reveal key properties about the function and its inverse, which can be useful in analyzing the security of the encryption method. Also, in physics, understanding inverse relationships helps in modeling phenomena where reversing the input and output makes sense, such as converting units or understanding reciprocal relationships between physical quantities.
The intersection point of a function and its inverse must lie on the line y = x . Given the points ( 0 , − 2 ) and ( 1 , − 1 ) , neither satisfies this condition, indicating they cannot be valid intersection points. Therefore, there must be a mistake in the problem statement or options provided.
;
