# Thread: Freaky function

1. ## Freaky function

Hi everyone.

y=f(x)=sqrt(1+(sqrt(1-(x^2-1)^2)))
I need to explain what is wrong but can not figure out nothing sensible. Also i need to say what does this function look like, in relation to the curve:
(x^2 - 1)^2 + (y^2 - 1)^2 = 1

Thx for any help

2. Originally Posted by Snowboarder
Hi everyone.

y=f(x)=sqrt(1+(sqrt(1-(x^2-1)^2)))
I need to explain what is wrong but can not figure out nothing sensible. Also i need to say what does this function look like, in relation to the curve:
(x^2 - 1)^2 + (y^2 - 1)^2 = 1

Thx for any help
$y = \sqrt{1 + \sqrt{1 - (x^2 - 1)^2}}$

Square both sides:

$y^2 = 1 + \sqrt{1 - (x^2 - 1)^2} \Rightarrow y^2 - 1 = \sqrt{x^2 - 1)^2}$.

Square both sides:

Over to you.

3. Originally Posted by Snowboarder
Hi everyone.

y=f(x)=sqrt(1+(sqrt(1-(x^2-1)^2)))
I need to explain what is wrong but can not figure out nothing sensible. Also i need to say what does this function look like, in relation to the curve:
(x^2 - 1)^2 + (y^2 - 1)^2 = 1

Thx for any help
Note: In the graph below I have translated the bottom relation three units down so you can see how the two graphs are related. In reality the top portions of these overlap.

I presume the question is why doesn't the first equation give the same graph as the second relation? Notice that the domain of the first function is restricted. So we will only get part of the relation below.

Actually, you can get the different pieces of the function that build the second relation:
$f(x) = \pm \sqrt{1+( \pm \sqrt{1-(x^2-1)^2})}$
where the $\pm$'s have no relation to each other. Notice as well if you manipulate the top expression and remove the square roots, you will get the relation on the bottom. The reason that this gives extra structure is that when you square both sides of an equation you add more solutions to your equation.

And hey, that's a cool graph.

-Dan