What is ^2? What type of function is that?

Use the mathematical meaning of bijective.

I do not want to get in to the whole definition of a function from a set-theoretic viewpoint, but the basic thing you need to show 2 things.

Step 1:One to OneShow that if and only if

Step 2:SurjectiveShow that for any (any real number) the equation has a solution.

Sometimes showing part 2 is more difficult, because you prove existence for that part. Sometime what you can do is prove that the function is countinous, then by the intermediate value theorem it must have a solution.

Of course, if you need to show that a function is bijective in a disk then do the same steps but only limited to that disk.