# Inverse function lingo

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

HallsofIvy
Homework Helper
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.

• 1 person
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
Staff Emeritus
Gold Member
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)

• 1 person
Mark44
Mentor
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.

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HallsofIvy