Question

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

Answer 1

Hint: To check the symmetry of the function we will check if it is symmetric about x axis, y – axis, origin and the line y = x. Now if the function is symmetric about y axis then we get the original equation after replacing x with – x. Similarly if the function is symmetric about the x axis then we get the original equation after replacing y with – y. Now if the function is symmetric about origin then we get the original equation after replacing x and y with – x and – y. If we interchange x and y and still obtain the original equation then the function is said to be symmetric around the line y = x.

Complete answer:
Let us first check symmetry with respect to y axis.
Now we have y = - x
To check the symmetry with respect to y axis replace x by – x. If the resultant equation is the same as the original equation then the function y is said to be symmetric with respect to y axis.
Now let us put – x in place of x.
$\begin{align}
  & \Rightarrow y=-\left( -x \right)=x \\
 & \Rightarrow y=x \\
\end{align}$
Since the equation is not similar to the original equation the function is not symmetric with respect to y axis.
Now let us check if the function is symmetric about the x axis.
Let us replace y by – y.
If we get the same equation then the function is symmetric about the x axis.
Let us put y = - y in the equation y = - x.
$\begin{align}
  & \Rightarrow -y=-x \\
 & \Rightarrow y=x \\
\end{align}$
Since we do not get the same equation the function is not symmetric about y axis.
Now let us check the symmetry about origin.
To check the symmetry about origin we will replace x by – x and y by – y.
If the obtained equation is the same as the original equation then the function is symmetric about origin.
Consider y = - x
Replacing x as – x and y as – y we get,
$\begin{align}
  & \Rightarrow -y=-\left( -x \right) \\
 & \Rightarrow -y=x \\
 & \Rightarrow y=-x \\
\end{align}$
Since we get the original equation back after simplifying we can say that the function is symmetric about origin.
Now we will check if the function is symmetric about the line y = x.
To check this symmetry we will simply interchange x and y and then if we get the original equation back after simplifying then the function is symmetric about line y = x.
The given equation is y = - x.
Now interchanging x and y we get,
$\begin{align}
  & \Rightarrow x=-y \\
 & \Rightarrow -x=y \\
 & \Rightarrow y=-x \\
\end{align}$
Since we get the original equation back we can say that the function is symmetric about y = x.
Hence the function is symmetric about origin and the line y = x.

Note: Note that if the function is symmetric about y axis then f(-x) = f(x) and the function is called an even function. Similarly if the function is symmetric about origin then we have f(-x) = -x
In this case the function is said to be an odd function. Also note that we can easily check if the function is symmetric by drawing a graph of the function. If the function does not change when rotated with respect to a line then the function is called symmetric with respect to that line.