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## Homework Statement

Sketch the graph of an invertible function f(x) that intersects its inverse in exactly three points.

## The Attempt at a Solution

I dont know where to start... Can I pick any function to start with?

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- Thread starter CrossFit415
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- #1

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Sketch the graph of an invertible function f(x) that intersects its inverse in exactly three points.

I dont know where to start... Can I pick any function to start with?

- #2

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You need to choose a function y=f(x) with the property that any two different values of x always yield two different values for y. Such functions are called injective functions and have graphs that are monotonic (i.e. either increasing or decreasing but not both).

Lastly, to find the inverse of such a graph, you need to reflect it in the line x=y (the 45 degree line). Try this with some examples and look at what all of the points of intersection between the function and its inverse have in common--that should help you to figure out how to get three points of intersection.

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- #4

HallsofIvy

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Yes, that's right. So in order to satisfy "intersects its inverse in exactly three points" there must be exactly three values of x such that f(x)= x. That is the same as saying f(x)- x= 0 for exactly three values of x. However the problem is making sure f is invertible then. You need to go from each such x to the next without having negative slope.

My first thought was to make f(x)- x a cubic. It's easy to construct a cubic that is 0 at three given values but a little harder to make sure it is always increasing. (Actually, there is a

- #5

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http://www.wolframalpha.com/input/?i=inverse+x^3

http://www.wolframalpha.com/input/?i=inverse+x^5

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