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Let $A= [{5}, \infty[$ , $B=[{1}, \infty[$ and $f: A \to B, f(x)=x^2-{10}x+{26}$

Determine the inverse image $f^{-1}:\ B \to A, f^{-1}(x)$

How should I proceed with this?

Thanks in advance

Derek Holt
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Emilycodes
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    Hint: $f(x)=(x-5)^2+1$. – ajotatxe Dec 08 '19 at 06:16
  • @mathmaniac. $f$ is most certainly surjective onto $[1,\infty)$. – Eric Dec 08 '19 at 06:20
  • @mathmaniac. https://en.wikipedia.org/wiki/Surjective_function – Eric Dec 08 '19 at 06:23
  • May be $x \mapsto \sqrt {x-1} + 5.$ – math maniac. Dec 08 '19 at 06:24
  • Solved, Thanks everyone! – Emilycodes Dec 08 '19 at 06:25
  • Sorry @Eric I only prove it for natural numbers. That's why the problem arises. I was trying to say that $f \big |_{A \cap \Bbb N}$ is not surjective. – math maniac. Dec 08 '19 at 06:26
  • @mathmaniac. I mean I guess that's true but I don't know why on earth you'd think to do that... – Eric Dec 08 '19 at 06:28
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    @Emilycodes you mean to ask for the inverse function, not the inverse image; see https://math.stackexchange.com/questions/3286150/understanding-the-difference-between-pre-image-and-inverse?rq=1. The inverse image/preimage of your function is $f^{-1}(B)=[5,\infty)$, the inverse function, given that $f$ is one-to-one on its domain, is $f^{-1}(x)=\sqrt{x-1}+5$. – Eric Dec 08 '19 at 06:34

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