Question

How do you determine if `y=sqrt(1-x^2)` is an even or odd function?

Answer 1

Easy method: graph it
Rigorous method: prove that it is odd or even using the definitions

Explanation:

Easy method: graph{y=sqrt(1-x^2) [-10, 10, -5, 5]}
By inspection, we see that the graph is symmetric about the y-axis so it is even.

Rigorous method:
We show that the function is even by showing that `f(x)=f(-x)` for all `x` in the domain, which in this case is `[-1,1] sub RR`. Thus, we get begin with `y=sqrt(1-x^2)` and we want to show that `y=sqrt(1-(-x)^2)` as well. This is easy for the problem since `(-x)^2=x^2` for real `x`, so clearly, the two are equal and the function is even.

We may also show that the function is not odd by showing that `f(x)=-f(-x)` for all `x` in the domain. Thus, we want to show that `sqrt(1-x^2)=-sqrt(1-(-x)^2)`. Clearly this is false as equality only holds when `sqrt(1-x^2)=0` or `x=+-1`, which clearly is not all `x`. Thus, the function is not odd.