Inverse Functions

Hello everyone.

We all know that the strict monotonicity(non-decreasing/non-increasing) of a function is a sufficient enough condition for it to have an inverse function.

Mathematically,

If f(x)f(x) is strictly monotonic,

then,

y=f(x)y=f(x) has an inverse function such that,

x=f1(y)x=f^{-1}(y)

Where, f1f^{-1} is the inverse of ff.

I started writing down the various functions whose inverse existed and proceeded to plot them on the same graph and invariably I found that the function and it's inverse were symmetric about y=xy=x.

Is this true for all functions whose inverse exists? Or just a few? If only few such functions exist, please tell me how to define such a class of functions.If all functions whose inverse exists show this property, then please post a proof for this, as I was unable find a valid proof with my current limited knowledge of math. You are free to use any mathematical analysis to prove it(just make it easy for me to understand).

I have placed this note into calculus assuming that calculus is required to prove this property. If you feel that it has to be placed under some other topic please let me know.

Note by Anirudh Chandramouli
3 years, 5 months 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](https://brilliant.org)example 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}

Comments

Sort by:

Top Newest

You probably don't need calculus.

  1. Suppose you have a invertible function f f . You don't know it's inverse, but you do know that f(a)=b f(a) = b for some a a and b b , i.e., the point (a,b) (a,b) lies on f f 's graph. Given this information, what point must exist on the inverse function's graph?

  2. Mathematically, what does it mean for two curves to be symmetric about a line? If like the first question, you are given a function but only know that (a,b) (a,b) lies on it's graph, what point must exist on the curve which is symmetric to it about the x=y x = y line.

Siddhartha Srivastava - 3 years, 5 months ago

Log in to reply

The function (1,3),(2,1),(3,2) (1,3), (2,1), (3,2) has an inverse and isn't monotonic.

Calvin Lin Staff - 3 years, 5 months ago

Log in to reply

I'm sorry I'm confused. I am unable get the analytical expression for the function you have just mentioned. And I rely completely in the expression to find the inverse. Please help me, Sir.

Anirudh Chandramouli - 3 years, 5 months ago

Log in to reply

The inverse function is

f1=(1,2),(2,3),(3,1) f^{-1} = (1,2), (2,3), (3,1) .

You never had the requirement that the domain of the function was a (continuous) interval.


Strict monotonicity is a sufficient, but not necessary, condition for an inverse to exist. The proof requires no Calculus at all, and it's just basic algebra.

Calvin Lin Staff - 3 years, 5 months ago

Log in to reply

@Calvin Lin Okay thank you sir. I got my mistake. I shall first change the topic to algebra. I think I have got a proof for it. Thank you

Anirudh Chandramouli - 3 years, 5 months ago

Log in to reply

Also I said that it is a sufficient condition for inverse to exist. Not that it is the only condition. I feel this is a fair argument. Please correct me if I'm wrong, sir

Anirudh Chandramouli - 3 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...