Question

Help me understand this question about irrational and complex numbers.

Answer 1

For what value of x does each expression represent a real number? \[\sqrt[3]{x-1}\]

Answer 2

@Loser66 @zepdrix

Answer 3

The answer is all reals, help me explain why

Answer 4

So what happens if we do this: \(\large\rm (-2)^3\)
? :)

Answer 5

-2*-2*-2=-8

Answer 6

So notice that an odd exponent will `preserve the sign` of our number.

Answer 7

Same with odd roots! :)
They don't carry the same restrictions that even roots do.

Answer 8

So you don't have the restriction of `only positive values under the root`,
like you would with an even root, like square root.

Answer 9

hmm.. I get what you are saying but how is it all roots?

Answer 10

\[\large\rm \sqrt[3]{-8}=-2\]If we have something like:\[\large\rm \sqrt[3]{x-1}\]You can at least clearly understand why x=-7 is ok, ya?

Answer 11

yup

Answer 12

How is it "all reals"? Is that what you meant?

Answer 13

yup

Answer 14

Because odd roots don't have the same restrictions as even roots :))\[\large\rm \sqrt[3]{-8}=-2\]\[\large\rm \sqrt{-8}=undefined~for~real~numbers\]
I'm not really sure how else to explain it.
We can't place negative numbers under a `square root`,
or any even numbered root for that matter.

So for even numbered roots, our domain is `all POSITIVE real numbers`.
That's what we're allowed to plug in.

For this odd root, we lose that restriction,
no values are restricted to x,
even the negatives are fair game.

So for the third root, our domain is `all real numbers.`

Answer 15

hmm I got it. SO does \[\sqrt[3]{4-x ^{2}}\] all reals as well?

Answer 16

Yes, because third root again :)

Answer 17

Gotcha! Thanks for the explanation and your time!!

Answer 18

nppp

Answer 19

You can find the inverse

\[\Large y = \sqrt[3]{x-1}\]
\[\Large x = \sqrt[3]{y-1}\]
\[\Large x^3 = y-1\]
\[\Large x^3+1 = y\]
\[\Large y = x^3+1\]


We can plug in any real number in for x, and get some real number out for y. So that proves the range of the original function is the set of all real numbers

Answer 20

@jim_thompson5910 Thanks :)

Answer 21

no problem