Question

for a function f(x) to have an inverse , than the functions derivative should be greater than 0, f(x)'>0, right?
so if f(x)=1/x, whose derivative is f(x)'= -1/x^2, than f(x) doesn't have an inverse

Answer 1

@Kainui

Answer 2

the derivative has to be monotone

Answer 3

do you remember that vertical stick test ?

Answer 4

it seems f(x) = 1/x passes that test right ?

Answer 5

Okay, I try.. :) Thanks.. :)

Answer 6

and include where the derivative does not exist?

Answer 7

the derivative has to be either f ' (x) >= 0 or f '(x) <= 0 on the entire interval

Answer 8

Honestly I am more a physicist/chemist so I will tell you that it has an inverse and that it is its own inverse since in all practicality it will give you exactly the inverse.

Just as a little bit of 'proof' look: \[\LARGE f(f^{-1}(x))=x\] this is the definition of an inverse as far as I'm concerned. \[\LARGE f(x)=\frac{1}{x}\] this is our function. \[\LARGE f(f(x))=\frac{1}{\frac{1}{x}}=x\]
Making the distinction here is sort of silly to me I guess. Mathematicians will have you do weird definitions and waste their time doing this pointless formalism where perhaps I doubt that this has an inverse when it seems as though it obviously does. lol

Answer 9

i thought u only use the horizontal line test..

Answer 10

you're right, its horizontal test... lets go thru Kai

Answer 11

ohk so what ur saying is that you can try to find the inverse, and than use f(f^−1(x))=x to check id its true?

Answer 12

i mean f(f^-1(x))=x

Answer 13

yes if its easy, however perl's statement works in general : `f'(x) doesn't change sign ==> f(x) has an inverse`

Answer 14

Yeah that was more opinion and me being critical about the current style that mathematics has since I don't find the concept of injective, surjective, bijective, sets, etc useful or appealing or meaningful to me at all. If at any point your function is multivalued you can always just define an inverse for each part depending on which is meaningful to you.

I can't see what you wrote @swift_13 it looks like a bunch of black question marks to me.

Answer 15

if f(x)=1/x and its inverse is also f(x)^-1=1/x

Answer 16

f(x) = 1/x

f'(x) = -1/x^2

which is always negative no matter what the x value is,
so the graph is always decreasing which clearly tells you that it passes the horizontal line test

Answer 17

and f(f(x)^-1)=x

Answer 18

if f(x) is a complicated function, you don't need to find the inverse to know whether an inverse exists or not

Answer 19

oh, ohk cause i was just going through my text book and it just mentions the f(x)' >0 to have an inverse

Answer 20

nothing about f(x)'<0

Answer 21

maybe take a screenshot and attach..

Answer 22

im not able to find a good sourse on this topic... but it makes sense intuituvely why the first derivative stuff works, right ?

Answer 23

If f ' (x) < 0 on the interval, then f is decreasing on the interval. If f is decreasing on the interval , then f is monotone. Then use this theorem I paste here.

https://proofwiki.org/wiki/Inverse_of_Strictly_Monotone_Function

(read the part about inverse)

Answer 24

yes it does make sense, so just to make sure if either the function is increasing or decreasing than it has an inverse

Answer 25

First time I'm hearing of a function being monotone, haha.

Answer 26

lol

Answer 27

so if f(x) = x^2 -2x

Answer 28

f(x)'=2x

Answer 29

f'(x) = 2x - 2 *

Answer 30

monotone functions have two possible states, they are either decreasing or they are increasing on the interval, but not both.

Answer 31

that cant have an inverse because the derivative changes from negative to positive

Answer 32

sorry yyea f(x)'=2x -2
thats means the function is both increasing and decreasing right

Answer 33

correct

Answer 34

so it can't have an inverse

Answer 35

thanks a lot!

Answer 36

no problem

Answer 37

also i kind of forgot but how do u check if the function is strictly increasing or decreasing or both

Answer 38

And the function itself does not pass the horizontal line test.

Answer 39

i mean its simple to c from f(x)'=2x -1 but for other functions

Answer 40

the only way to check for horizontal line test is to draw a graph right? another way?

Answer 41

you have derivative now

Answer 42

f(x) passes horizontal line test if :

f'(x) > 0 for all x

or

f'(x) < 0 for all x

Answer 43

its an exclusive or, not just or

Answer 44

Yeah, and because your derivative \(\ f'(x) = 2x - 2\) moves from negative values to positive values, it fails the test.

Answer 45

you could also try proving \(\large f(b) \gt f(a)\) whenever \(\large b \gt a\) or vice versa

Answer 46

above trick is more used in sequences though..

Answer 47

ohk thanks guys!

Answer 48

but like for f(x)'=2x-2 , would i say for f(x)' > 0 for all x>1 and f(x)'<0 for all x <1

Answer 49

due to this the function doesn't have an inverse

Answer 50

i mean due to this it doesn't pass the horizontal line test

Answer 51

f '(x) <0 for x<1

Answer 52

thats correct, the derivative has to stay positive or stay negative in order for the inverse of the original function to exist

Answer 53

yes.. however if you take only the piece x>1, then it surely has an inverse because that piece alone passes the horizontal line test.. but it would be like chaning the original problem... so you simply say : `the given function has no inverse because f'(x) changes signs`

Answer 54

yup thanks!

Answer 55

yup got it! i understand now!