Can a Non-Invertible Function Become Invertible in Certain Intervals?

[ZeroPivot]

can a function that's not inversable be inversible in certain interwalls. is it ok to say its inversable in this specific intervall or can't the function ever be called inversible?

 

[HallsofIvy]

Yes, but technically, it wouldn't be the same function.

For example, $f(x)= x^2$is not "invertible" because it is neither "one to one" nor "onto". However, if we restrict the domain to the "non-negative real numbers" then its inverse is $\sqrt{x}$. If we restrict the domain to the "non-positive real numbers" then its inverse is $-\sqrt{x}$.

However, the domain of a function is as much a part of its definition as the "formula". That is, "$f(x)= x^2$, for x any real number", "$g(x)=x^2$, for x any non-negative real number", and $h(x)= x^2$, for x any non-negative real number" are three different functions. The first does not have an inverse, the last two do.

 

[ZeroPivot]

Yes, but technically, it wouldn't be the same function.

For example, $f(x)= x^2$is not "invertible" because it is neither "one to one" nor "onto". However, if we restrict the domain to the "non-negative real numbers" then its inverse is $\sqrt{x}$. If we restrict the domain to the "non-positive real numbers" then its inverse is $-\sqrt{x}$.

However, the domain of a function is as much a part of its definition as the "formula". That is, "$f(x)= x^2$, for x any real number", "$g(x)=x^2$, for x any non-negative real number", and $h(x)= x^2$, for x any non-negative real number" are three different functions. The first does not have an inverse, the last two do.

can i say function y=x^2 is inversible for x E [0 ,4] ?

 

[Office_Shredder]

Yes, the function
$$f: [0,4]\to [0,16],\ x\mapsto x^2$$
is an invertible function because of the domain that has been specified (also it needs to have the right codomain, or it won't be an onto function, but that's more of a technicality that is washed away by restricting the codomain to whatever the range is)

 

[Mark44]

can i say function y=x^2 is inversible for x E [0 ,4] ?

"Inversible" is not a word - the one you want is invertible.

 

[HallsofIvy]

So why is "reversible" a word and not "revertible"?

 

[Mark44]

"Inversible" is not a word - the one you want is invertible.

Possibly because the adjectives invertible and reversible come from the verbs to invert and to reverse, respectively.