Question

Not able to get it :|
x^12 - x^9 + x^4 - x + 1 ---> Find DOMAIN.

Answer 1

Incorrectly typed.
Its,
The same expression in a sqrt.

Answer 2

@siddhantsharan Is there any value of x which will make
\[x^{12}-x^9+x^4+1\ \text{undefined}??\]

Answer 3

Yep. When its negative.

Answer 4

So what x can't be or should be?

Answer 5

How do I figure out when is this expression negative?

Answer 6

Basically the condition imposed should be,
x^12 - x^9 + x^4 - x + 1 > 0

Answer 7

Yeah you're right, l'm also thinking

Answer 8

@experimentX Help?

Answer 9

lol ... seems factorization .. huge factorizaton

Answer 10

Yeah. It can be written as
x(x^8 +1)(x^3 - 1) + 1 > 0
Intuitively that does help as we can see that its valid for all R.
But how do you prove it mathematically? Or how does the factorization help.

Answer 11

from here, x >= 1 or x<= -1

Answer 12

How?

Answer 13

x^8 is always +ve,
so, x(x^3-1) must always be +ve

Answer 14

lol, x<0 also satisfies condition

Answer 15

No. Not necessarily. There is a + 1 too. That makes a lot of diff here.

Answer 16

i guess (0,1) must be excluded

Answer 17

You cant say that.
If x < 0 Then it is always satisffied.
And check for 1/2.
Its still satisfying.

Answer 18

does not satisfy..

Answer 19

Does.

Answer 20

It gives 0.53 approximately.

Answer 21

let's graph x(x^3-1)
http://www.wolframalpha.com/input/?i=x%28x^3-1%29

Answer 22

No, graph x(x^3 - 1)(x^ 8 + 1) + 1.
You cant ignore the rest of the terms as they matter a lot for x between 0 to 1.

Answer 23

http://www.wolframalpha.com/input/?i=x%28x%5E3+-+1%29%28x%5E+8+%2B+1%29++%2B+1

Answer 24

You get me right?

Answer 25

Oh ... sorry,

Answer 26

lol. :)

Answer 27

x(x^8+1)(x^3-1) > -1

Answer 28

Yep.

Answer 29

it implies,
x(x^3 -1) > -1/(x^8+1)

It seems it satisfies this property for all values of x

Answer 30

as x->0, x(x^3 -1)->0 > -1

as x<0, x(x^3 -1>0 > -1

for, x>1, x(x^3 -1)>0>-1


the only place we need to check is (0,1)
the minimum value is http://www.wolframalpha.com/input/?i=minima+x%28x^3+-+1%29%28x^+8+%2B+1%29 > -1

So the domain is all real numbers