If $f(x)$ and its inverse function, $f^{-1}(x)$, are both plotted on the same coordinate plane, what is their point of intersection?
Answer (2)
The intersection point of a function f ( x ) and its inverse f − 1 ( x ) lies on the line y = x .
To find the intersection, solve the equation f ( x ) = x .
The solution x gives the intersection point ( x , x ) .
Therefore, the point of intersection is ( x , x ) , where x is the solution to f ( x ) = x .
Explanation
Understanding the Intersection When a function, f ( x ) , and its inverse, f − 1 ( x ) , are plotted on the same coordinate plane, their point(s) of intersection lie on the line y = x . This is because the inverse function is a reflection of the original function over the line y = x .
Setting up the Equation To find the point of intersection, we need to solve the equation f ( x ) = x . The solution to this equation will give us the x-coordinate of the intersection point. The y-coordinate will be the same as the x-coordinate since the intersection point lies on the line y = x .
Finding the Intersection Point Thus, the point of intersection is ( x , x ) , where x is the solution to f ( x ) = x .
Examples
Consider a scenario where you are tracking the conversion rate of leads to customers for your business. If f ( x ) represents the number of customers you get from x leads, then f − 1 ( x ) would represent the number of leads needed to get x customers. The point where f ( x ) and f − 1 ( x ) intersect on a graph (i.e., where f ( x ) = x ) indicates a breakeven point where the number of leads equals the number of customers. This helps in understanding the efficiency of your conversion process and setting realistic targets.
The intersection point of a function f ( x ) and its inverse f − 1 ( x ) occurs where they are equal, or on the line y = x . This is found by solving the equation f ( x ) = x , which gives the coordinates of the intersection as ( x , x ) . Therefore, the intersection point is ( x , x ) , where x is the solution to the equation.
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