Question

is it true that only bijective functions are invertible?

Answer 1

i think so only bijective functions pass horizontal line test

Answer 2

how can bijective *functions* pass horizontal line test??

Answer 3

its a one-to-one, so y values are not repeated

Answer 4

yes

Answer 5

but a bijective requires both 1/1 + onto, not sure if onto plays a role in inverse..

Answer 6

im just asking about the inverse....

Answer 7

for a function to be invertible, it requires to be one-to-one

but a bijective function function is both one-to-one and onto. so the given statement is false, becoz some other functions that are not onto are also invertible

Answer 8

@ganeshie8 I think it does; If your function is not onto, then there might be elements in the co-domain which have no pre-images, consequently, your apparent inverse function will have pre-images with no images,

Answer 9

agree with zugzwang...

Answer 10

the inverse function is defined based on range or codomain ?

Answer 11

i think what @ganeshie8 is the one-to-one correspondence (which is another term for bijection)

Answer 12

if it is range, the function doesnt require onto ness for it it be invertible, just the one-to-one correspondence is sufficient

Answer 13

If your function is onto, your range and codomain are equal.

Answer 14

bijection = one-to-one + onto

Answer 15

A bijection (or bijective function or one-to-one correspondence) is a function giving an exact pairing of the elements of two sets.

~ http://en.wikipedia.org/wiki/Bijection

Answer 16

search onto in the same wiki page

Answer 17

Let
f: X --> Y
be an invertible function. Then let f*: Y --> X be its inverse.

Suppose f is not one to one.
Then there exists x1, x2 in x such that x1 is not equal to x2 and
f(x1) = f(x2) = y
Then
f*(y) = x1 since y = f(x1)
and
f*(y) = x2 since y = f(x2)
Then f* is not well defined, since x1 is not equal to x2
thus, f must be one to one

Suppose f is not onto
Then there exists y in Y such that for all x in X, f(x) is not y.
Then f*(y) does not exist, since there is not x in X such that y = f(x)
Thus f* is not defined on its entire domain Y, meaning it is not a function, which is a contradiction. Thus, f must be onto

Then f is bijective

Answer 18

so a function requires onto to be invertible ?

Answer 19

requires onto AND one-to-one

Answer 20

makes sense :)

Answer 21

Something like this was in an exam I took just two weeks ago >.>
If fate is kind to me, I'll never have to deal with relations again O.o