Objectives: In this tutorial, we define what it means for a function to be symmetric with respect to the y-axis and the origin. Algebraic conditions for these two types of symmetry are obtained from the geometrical definitions. Some examples of functions illustrate these different symmetries. After working through these materials, the student should be able

• to recognize from the graph of a function whether a function is even, odd or neither; and
• to show algebraically whether a function is even, odd or neither.

Modules:

 DEFINITION. A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f. A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.
• Discussion [Using Flash]

• Examples.

• Comment: From the previous MathView notebook, we see that a function f is odd if and only if the graphs of y = f(x) and y = -f(-x) coincide. Similarly, a function f is even if and only if the graphs of y = f(x) and y = f(-x) coincide.

• A Quiz on whether a given function is even or odd.