Inverse function lingo

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  • #1
ZeroPivot
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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?
 

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  • #2
HallsofIvy
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Yes, but technically, it wouldn't be the same function.

For example, [itex]f(x)= x^2[/itex]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 [itex]\sqrt{x}[/itex]. If we restrict the domain to the "non-positive real numbers" then its inverse is [itex]-\sqrt{x}[/itex].

However, the domain of a function is as much a part of its definition as the "formula". That is, "[itex]f(x)= x^2[/itex], for x any real number", "[itex]g(x)=x^2[/itex], for x any non-negative real number", and [itex]h(x)= x^2[/itex], for x any non-negative real number" are three different functions. The first does not have an inverse, the last two do.
 
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  • #3
ZeroPivot
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Yes, but technically, it wouldn't be the same function.

For example, [itex]f(x)= x^2[/itex]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 [itex]\sqrt{x}[/itex]. If we restrict the domain to the "non-positive real numbers" then its inverse is [itex]-\sqrt{x}[/itex].

However, the domain of a function is as much a part of its definition as the "formula". That is, "[itex]f(x)= x^2[/itex], for x any real number", "[itex]g(x)=x^2[/itex], for x any non-negative real number", and [itex]h(x)= x^2[/itex], 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] ?
 
  • #4
Office_Shredder
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Yes, the function
[tex] f: [0,4]\to [0,16],\ x\mapsto x^2 [/tex]
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)
 
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  • #5
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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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  • #6
HallsofIvy
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So why is "reversible" a word and not "revertible"?
 
  • #7
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"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.
 

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