Even and Odd Functions
A Function can be classified as Even, Odd or Neither. This classification can be
determined graphically or algebraically.
Graphical Interpretation -
Even Functions: Odd Functions:
Have a graph that is Have a graph that is
symmetric with respect symmetric with respect
to the Y-Axis. to the Origin.
Algebraic Test โ Substitute (โ๐ฅ) in for ๐ฅ everywhere in the function and analyze the
results of ๐(โ๐ฅ), by comparing it to the original function ๐(๐ฅ).
Even Function: ๐ = ๐(๐) is Even when, for each ๐ฅ in the domain of
๐(๐ฅ), ๐(โ๐ฅ)= ๐(๐ฅ)
Odd Function: ๐ = ๐(๐) is Odd when, for each ๐ฅ in the domain of
๐(๐ฅ), ๐(โ๐ฅ)= โ๐(๐ฅ)
Examples:
a. ๐(๐)= ๐๐+ ๐ b. ๐(๐)= ๐๐โ๐๐ c. ๐(๐)= ๐๐โ๐๐ + ๐
๐(โ๐)=(โ๐)๐+ ๐ ๐(โ๐)=(โ๐)๐โ ๐(โ๐) ๐(๐)=(โ๐)๐โ ๐(โ๐)+ ๐
๐(โ๐)= ๐๐+ ๐ ๐(โ๐)= โ๐๐+๐๐ ๐(โ๐)= ๐๐+๐๐ + ๐
๐(โ๐)= ๐(๐) ๐(โ๐)= โ(๐๐โ๐๐)= โ๐(๐) ๐(โ๐)โ ๐(๐)โ โ๐(๐)
Y-Axis โ acts like a mirror
Origin โ If you spin the picture upside down
about the Origin, the graph looks the same!
Origin
Even Function!
Odd Function!
Neither!
