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#functions
cursive grove
slender haloBOT
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dark relic
Find a function f that is injective but not surjective
@cursive grove
You will be able to construct a g such that g o f=I_d
cursive grove
Find a function f that is injective but not surjective
I’m just start learning calculus so don’t really know what you mean
Or do I supposed to know these words?
dark relic
Do you know what a one to one function is ?
cursive grove
Do you know what a one to one function is ?
Every x corresponds to a unique f(x)
dark relic
Every x corresponds to a unique f(x)
Yeah, now try to find a one to one function f from R to R that isn’t onto you will be able to construct a left inverse (a function g such that g o f=I)
But it won’t have a right inverse
cursive grove
Yeah, now try to find a one to one function f from R to R that isn’t onto you wi...
So I looked up wiki what a subjective(or onto) function is. So basically let f(x)=x so that f is one to one, but when x is 0 f(x) is not defined so that f is not onto?
dark relic
What about f(0)=0, and other than that f is not defined?
f needs to be defined for all x and needs to be one to one, but it’s image mustn’t be all of R
Take for example a one to one function that only has positive images
cursive grove
@dark relic Is image a set of f(x)?
dark relic
<@757307393347551292> Is image a set of f(x)?
When I said images I meant if x is a real number then f(x) is positive
cursive grove
When I said images I meant if x is a real number then f(x) is positive
So a image is a statement?
dark relic
Yeah
I may not have expressed myself that well but I hope you understand what I mean now
cursive grove
I may not have expressed myself that well but I hope you understand what I mean ...
But what I searched was : the image of x is f at x. Instead of a if-then statement
dark relic
But what I searched was : the image of x is f at x. Instead of a if-then stateme...
Do you mean f(x)=x for all x ?
Like I said if you want g o f=I (I is the identity function) then f must be defined for any x
It can’t be defined for only certain points
cursive grove
Do you mean f(x)=x for all x ?
No. I mean like the image of 5 is f(5)
cursive grove
Oh so you mean f(x)=x for a certain x
No
dark relic
Oh okay I misunderstood my bad
cursive grove
I’m not saying they are equal
dark relic
You literally mean that the image of x is f(x)
Yes indeed
The idea is that is f is a one to one but not onto function then you will be able to to construct a left inverse g by for example setting g(y)=x if y=f(x) for some x in R and g(y)=y if y is not in the image of f (ie there is no x such that y=f(x), which exists because f is not onto)
In that case you indeed have g o f=I but there will be no h such that f o h=I
dark relic
The idea is that is f is a one to one but not onto function then you will be abl...
g Will be well defined considering f is one to one
try to explicitly find an example
cursive grove
Take for example a one to one function that only has positive images
If a one to one function has only positive image, wouldn’t happen that if x is negative, then the value of f(x) must be positive, and therefore overlap with positive number y, which means f is not one to one?
dark relic
take f(x)=e^x, it’s positive for all x it’s one to one but not onto
cursive grove
Indeed. Thanks for helping.
dark relic
Indeed. Thanks for helping.
Np, from f(x)=e^x you can construct g
gray grove
F(x) = 0
hazy swiftBOT
@cursive grove
cursive grove
+close