Inverse Functions and their intersection points

Say you have $f(x)$ and $g(x)$ and $g(x) = f^{-1}(x)$ .

I observed that these two curves need not intersect, for example with $f(x) = e^x$ and $g(x) = \ln x$ never intersecting each other.

I also observed that a function can either have one, two, or three intersections with its inverse, but I was unable to find a function which has more than 3 intersection points with its inverse.

How would I prove or disprove the hypothesis that an elementary function and its intersection can only have up to 3 intersection points? Any counterexamples are appreciated!

#Algebra

Easy Math Editor

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Comments

Consider $f(x) = x$ .

Note that $g(x) = f^{-1}(x) = x = f(x)$ .

So, there are infinitely many intersections!

A Former Brilliant Member - 4 years, 10 months ago

Good solution. I am aware of the infinite intersections solution, but does anyone have any functions which have 4 or more intersections with their inverse (but not an infinite number of intersections)?

Oli Hohman - 4 years, 10 months ago

Consider, over any finite interval say $X$ , $f(x) = x + \sin x$ .

Over the interval $X$ there are finitely many intersections. The exact number depends on $X$ itself. But you can have any finite number of intersections.

A Former Brilliant Member - 4 years, 10 months ago

Nice solution

Ian Limarta - 4 years, 10 months ago

At first, I was unsure of how to find the inverse of that function, so I decided to graph it to verify your claim and you're right!

http://www.wolframalpha.com/input/?i=find+inverse+of+f(x)+%3D+x%2Bsin(x)

Oli Hohman - 4 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$