Question

How do you test for symmetry with respect to the line y=-x?

Answer 1

This function is symmetric with respect to the origin.

Explanation:

We have to test whether the function `y=-x` is even or odd.
When a function is even , it is symmetric with respect to the y-axis.
When a function is odd, it is symmetric with respect to the origin.
A function is even if `f(-x)=f(x)`
A function is odd if `f(-x)=-f(x)`
A function could be neither odd nor even.
In this case, we replace `y` with `f(x)`
We ask ourselves:
Is this function even?
`f(-x)=x` and that is not equal to `-x`.
Therefore, no.

We ask ourselves:
Is this function odd?
`f(-x)=x` and that is equal to `-f(x)=x`.
Therefore, yes.

We now know that this function is odd, meaning that this function is symmetric with respect to the origin.