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Summary of even and odd functions
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.
From the definition itself, we see how can we test if the functions are even or odd.
- First graph (to the left)
From the graph, we see that the function is symmetric with respect to the y-axis because the left side (x<0) is a reflection (along the y-axis) of the right side (x>0), so it is an even function.
- Second graph (to the right)
From the graph, we see that the function is symmetric with respect to the origin, the left side (x<0) is a reflection (along with the origin) of the side to the right (x>0), so it is an odd function.
- First function (to the left)
The function:
is an even function because we see that:
- Second function (to the right)
The function:
is an odd function because we see that:
Summary
We have the following results:
- First function (to the left): even
- Second graph (to the right): odd
- First function (to the left): even
- Second function (to the right): odd