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I need to determine the inverse function for the following function:

$$ f:\mathbb{R}\to\mathbb{R}\ \ \ \text{with}\ \ f(x)=x^2\\ \text{so I need to determine}\\ f^{-1}(\{25\}) $$

So I know that the function $f(x)=x^2$ is bijective, so it has a inverse.

I was thinking to do the assignment, in the following manner: $$ \text{find the inverse of $f(x)=x^2$}\\ y=x^2\\ \pm\sqrt{y}=x\\ f^{-1}(x)=\pm\sqrt{x}\\ \text{now that I have the inverse function, I can put $25$ inside, so I get:}\\ f^{-1}(\{25\})=\pm5\\ $$

Did I do this right? Any advice appreciated.

Thank you!

egreg
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depecheSoul
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    The function has no inverse, because it's not injective. – egreg Dec 26 '13 at 14:32
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    I believe you want inverse image (or preimage); nothing to do with inverse functions! – Ted Shifrin Dec 26 '13 at 14:37
  • As the OP shows, he wants the inverse image as the argument to $f^{-1}$ is a set and not a number. Just imprecise use of language but his mathematical notation is quite precise. – user44197 Dec 26 '13 at 14:38

2 Answers2

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Yes, except your teacher may want you to write the answer out in a set form, i.e $$ f^{-1}({25}) = \{5, -5\} $$ $\pm 5$ is just a shorthand when writing out solutions, not sets.

user44197
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Your function is not bijective (f(-5)=f(5)=25). Considere the induced function $R^+ \leftarrow R^+$ which is bijective

John
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