Chapter 12.5, Problem 41PPSE

Solution

To find: To explain why a reflection of a points across $y=x$ is equivalent to finding the inverse of the function.

The reflection across the line $y=x$ is equivalent to the inverse of the function.

Given information:

Given, to explain why a reflection of a points across $y=x$ is equivalent to finding the inverse of the function.

Calculation:

A function of the form $f\left(x\right)=y$ contains the point $\left(x,y\right)$ and its inverse is given by $f^{-1}\left(y\right)=x$ contains the point $\left(y,x\right)$

When a point is reflected across the line $y=x$ , the points of the form $\left(x,y\right)$ transform to $\left(y,x\right)$

It can be observed that the matrix and the original function have same points, while the reflection and the inverse function have same points.

Conclusion:

Hence, the reflection across the line $y=x$ is equivalent to the inverse of the function.