### Operations for functions: New functions from old

### Symmetry of functions

Symmetry of functions is more of a geometrical concept than an algebraic concept. A function can be symmetric about a vertical line or about a point. However, a function cannot be symmetric about the #x#-axis (or any other horizontal line of the type \(y=a\)), where \(a\) is a constant), since anything that is mirrored around an horizontal line will not pass the *vertical line test *(and for that reason cannot be a function by definition). Two special cases are even and odd functions.

Symmetry of a graph

A graph is said to be **symmetric with respect to a vertical line** #\ell# if, for each point of the graph, its image under reflection about #\ell# is also a point of the graph. The line we will mostly be looking at is the #y#-axis.

A graph is said to be **symmetric with respect to a point** #P# if, for each point of the graph, its image under the point reflection about #P# is also a point of the graph. The point we will mostly be looking at is the origin.

A circle is symmetric with respect to its center and with respect to each line through the center.

A line is symmetric with respect to each point on the line.

Even function

A function \(f\) is said to be**even**if its graph is symmetric with respect to the #y#-axis. In other words: if \[f(-x)=f(x)\] for each point #x# in the domain of #f#.

For instance, the function #f(x)=x^2# is even:

Odd function

A function \(f\) is called an **odd function** if its graph is symmetric with respect to the origin. It has the property \[f(-x)=-f(x)\]

For instance, the function #f(x)=x^3# is odd:

This can be seen in the following graph of the function #f#:

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