If $f(x)$ and its inverse function, $f^{-1}(x)$, are both plotted on the same coordinate plane, where is their point of intersection?
A. $(0,6)$
B. $(1,4)$
C. $(2,2)$
D. $(3,0)
Answer (2)
The point of intersection of a function and its inverse lies on the line y = x .
Check each option to see if the x and y coordinates are equal.
The point (2,2) is the only option where the x and y coordinates are equal.
Therefore, the point of intersection is ( 2 , 2 ) .
Explanation
Understanding Inverse Functions When a function and its inverse are plotted on the same coordinate plane, their point of intersection must lie on the line y = x . This is because the inverse function is a reflection of the original function across the line y = x . Therefore, the coordinates of the intersection point must be equal.
Checking the Options Now, let's examine the given options:
(0,6): The x and y coordinates are not equal.
(1,4): The x and y coordinates are not equal.
(2,2): The x and y coordinates are equal.
(3,0): The x and y coordinates are not equal.
Determining the Intersection Point Only the point (2,2) has equal x and y coordinates. Therefore, the point of intersection of f ( x ) and f − 1 ( x ) is (2,2).
Examples
Imagine you're looking at a map where one axis represents the time it takes to travel from city A to city B, and the other axis represents the distance. The inverse function would represent the time it takes to travel from city B to city A for a given distance. The point where these two functions intersect (if they do) would be a specific distance and time where the travel time is the same in both directions. This concept is useful in logistics, route planning, and understanding symmetrical relationships in various real-world scenarios.
The point of intersection between a function and its inverse can be found where the coordinates are equal. In this case, the only option where x equals y is (2,2), making it the intersection point. Therefore, the answer is (2,2).
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