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 $f(x)$ is strictly monotonic,

then,

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

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

Where, $f^{-1}$ is the inverse of $f$ .

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 $y=x$ .

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.

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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}`	$\boxed{123}$

Comments

You probably don't need calculus.

Suppose you have a invertible function $f$ . You don't know it's inverse, but you do know that $f(a) = b$ for some $a$ and $b$ , i.e., the point $(a,b)$ lies on $f$ 's graph. Given this information, what point must exist on the inverse function's graph?
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 $(a,b)$ lies on it's graph, what point must exist on the curve which is symmetric to it about the $x = y$ line.

Siddhartha Srivastava - 5 years, 2 months ago

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

Calvin Lin Staff - 5 years, 2 months ago

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 - 5 years, 2 months ago

The inverse function is

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

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 – 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

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

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}`	$\boxed{123}$