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

1 Answer
Dec 2, 2017

Answer:

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.