### Video Transcript

Shown is the graph of π of π₯ equals five π₯ cubed plus six. Find the intersection of the inverse function π inverse of π₯ with the π₯-axis.

We recall first that if a function π is invertible, then the graph of its inverse function is found by reflecting the graph of the original function in the line π¦ equals π₯. In doing so, the roles of π₯ and π¦ in the function are interchanged so that we have π₯ equals π of π¦, which is equivalent to π¦ equals π inverse of π₯. This means that a point with coordinates π, π on the graph of the original function π¦ equals π of π₯ will be mapped to a point with coordinates π, π on the graph of the inverse function π¦ equals π inverse of π₯.

Weβre looking for the intersection of the inverse function with the π₯-axis. And we know that everywhere on the π₯-axis, π¦ is equal to zero. So weβre looking for a point or points with coordinates π, zero. But from what weβve just discussed about the relationship between the graphs of π¦ equals π of π₯ and π¦ equals π inverse of π₯, this point will have been mapped from a point with coordinates zero, π on the graph of π¦ equals π of π₯. As the π₯-coordinate of this point is zero, this is a point on the π¦-axis. So in other words, we can determine the π₯-intercept of the graph of π¦ equals π inverse of π₯ by considering the π¦-intercept of the graph of π¦ equals π of π₯.

From the figure weβve been given, we see that the graph of π¦ equals π of π₯ passes through the π¦-axis at a value of six. In other words, the coordinates of its π¦-intercept are zero, six. And this will be mapped to the point six, zero for the graph of the inverse function. So this is the point of intersection of the inverse function with the π₯-axis. We can also sketch the graph of the inverse function π¦ equals π inverse of π₯ by reflecting π¦ equals π of π₯ in the line π¦ equals π₯ and confirm that it does indeed intersect the π₯-axis at the point six, zero.