Inverse Functions and their intersection points

Say you have f(x)f(x) and g(x)g(x) and g(x)=f1(x)g(x) = f^{-1}(x) .

I observed that these two curves need not intersect, for example with f(x)=exf(x) = e^x and g(x)=lnxg(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!

Note by Oli Hohman
5 years ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

Consider f(x)=xf(x) = x.

Note that g(x)=f1(x)=x=f(x)g(x) = f^{-1}(x) = x = f(x).

So, there are infinitely many intersections!

Log in to reply

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 - 5 years ago

Log in to reply

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

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

Log in to reply

Nice solution

Ian Limarta - 4 years, 12 months ago

Log in to reply

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!

Oli Hohman - 4 years, 8 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...