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

Question

Serenity Johnson · Accepted Answer

This function is symmetric with respect to the origin.

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.

Liam Brennan · Answer

undefined To test for symmetry with respect to the line y = -x, you can use the following method:

1. Substitute y = -x into the equation of the given function.
2. Simplify the resulting expression.
3. If the simplified expression is equivalent to the original function, then the function is symmetric with respect to the line y = -x. If not, then the function is not symmetric with respect to this line.

This test essentially involves replacing every occurrence of y in the equation of the function with -x, and then checking if the resulting expression remains unchanged or not. If it does, then the function exhibits symmetry with respect to the line y = -x.