#inverse functions

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dusk citrus
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could someone please explain to me how to find the correct domains and such for these questions and tell me where i went wrong

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flat nebulaBOT
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viscid matrix
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When is sqrt(x) a real number?

dusk citrus
viscid matrix
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Do you not know what a real number is?

dusk citrus
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yes

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i think so

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like anything that isnt a square root right

viscid matrix
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No.

dusk citrus
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oh

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is it smth similar

viscid matrix
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Probably not?

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I don't know what would be "similar" in your mind.

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But there are definitely square roots that are real.

dusk citrus
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what would u classify a real number then

viscid matrix
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I mean, the definition is probably way beyond the scope here.

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What's important is that real numbers are (isomorphic to) complex numbers with imaginary part 0.

dusk citrus
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this is really hard i dont even remember learning this

viscid matrix
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Okay, look.

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Pick a real number. Any real number.

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Now square it.

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What can we say about the result?

dusk citrus
dusk citrus
viscid matrix
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Look. What's a negative times a negative?

dusk citrus
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positive

viscid matrix
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What's a positive times a positive?

dusk citrus
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positive also

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is a real number just positive

viscid matrix
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No.

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What's zero times zero?

dusk citrus
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0

viscid matrix
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So all real numbers are either negative or positive or 0.

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So for any real number x, x^2 is always positive or 0.

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In particular, it's never negative.

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So if y = x^2, and x is a real number, then y is nonnegative.

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And if y = x^2, then x = sqrt(y).

dusk citrus
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OH sqrt is squareroot

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why did i not understand that

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ok i get that bit nowi think

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so how do i find thr domain knowing this

viscid matrix
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Well, when is x not a real number?

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Or, wait.

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We said that any real number squared is nonnegative, right?

dusk citrus
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yes

viscid matrix
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So then the question is, what kind of number squared would be negative?

dusk citrus
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a non real number?

viscid matrix
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Right. In particular, we call them imaginary numbers, and the sum of a real and imaginary number is a complex number.

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But to take it back to your problem, we want the inverse function to only output real numbers.

dusk citrus
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okok i understand

viscid matrix
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So what needs to happen for that?

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When is sqrt(x) a real number?

dusk citrus
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when its greater than0 maybe

viscid matrix
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"It"?

dusk citrus
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x

viscid matrix
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When x is nonnegative, then sqrt(x) is real.

dusk citrus
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ohhhhh

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does that include 0

viscid matrix
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Yes.

dusk citrus
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or no

viscid matrix
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sqrt(x) is real if and only if x >= 0.

dusk citrus
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okok im understanding

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how come for 5 then the domain is greater than 5

viscid matrix
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Because the question explicitly states that the domain is real numbers greater than some a.

dusk citrus
dusk citrus
viscid matrix
dusk citrus
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they are opposite with the original function right

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it gets swapped

viscid matrix
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Okay, so a function is defined as a mapping from a set A, the domain, to a set B, the codomain, such that every element of A is mapped to exactly one element of B.

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We call a function an injection, or injective, if every element of A maps to a distinct element of B.

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That is, x =/= y implies f(x) =/= f(y).

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We call a function a surjection, or surjective, if every element of B is mapped to by some element of A.

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That is, for all y in B, there exists x in A such that f(x) = y.

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Am I making sense so far, @dusk citrus?

dusk citrus
hoary nicheBOT
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@dusk citrus

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dusk citrus
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+close

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