Introduction to Even and Odd Functions
A graph is symmetric if one half looks like the other half. We might also say that one half of the graph is a reflection of the other.
Symmetric Graphs
When a graph has symmetry, we usually say that it is symmetric with respect to a line or a point. The graph in Figure 3.15 is symmetric with respect to the x axis because the half of the graph above the x axis is a reflection of the half below the x axis. The graph in Figure 3.16 is symmetric with respect to the y axis.
The graph in Figure 3.17 is symmetric with respect to the vertical line x = 2.
Origin of Symmetry
One type of symmetry that is a little harder to see is origin symmetry . A graph has origin symmetry if folding the graph along the x axis then again along the y axis would have one part of the graph coincide with the other part. The graphs in Figures 3.18 and 3.19 have origin symmetry.
Knowing in advance whether or not the graph of a function is symmetric can make sketching the graph less work. We can use algebra to decide if the graph of a function has y axis symmetry or origin symmetry. Except for the function f ( x ) = 0, the graph of a function will not have x axis symmetry because x axis symmetry would cause a graph to fail the Vertical Line Test.
For the graph of a function to be symmetric with respect to the y axis, a point on the left side of the y axis will have a mirror image on the right side of the graph.
Even and Odd Functions
Even Functions
The graph of a function with y axis symmetry has the property that (x, y) is on the graph means that (− x, y ) is also on the graph. The functional notation for this idea is f(x) = f (− x). “ f(x) = f (− x)” says that the y value for x (f(x)) is the same as the y value for − x (f (− x)). If evaluating a function at − x does not change the equation, then its graph will have yaxis symmetry. Such functions are called even functions.
For a function whose graph is symmetric with respect to the origin, the mirror image of ( x , y ) is (− x, − y ).
Odd Functions
The functional notation for this idea is f (− x ) = − f ( x ). “ f (− x ) = − f ( x )” says that the y value for − x ( f (− x )) is the opposite of the y value for x (− f ( x )). If evaluating a function at − x changes the equation to its negative, then the graph of the function will be symmetric with respect to the origin. These functions are called odd functions .
In order to work the following problems, we will need the following facts.
a (− x ) ^{even power} = ax ^{even power} and a (− x ) ^{odd power} = − ax ^{odd power}
Examples
Determine if the given function is even (its graph is symmetric with respect to the y axis), odd (its graph is symmetric with respect to the origin), or neither.

f ( x ) = x ^{2} − 2
Does evaluating f(x) at − x change the function? If so, is f (− x ) = −( x ^{2} − 2)= − f ( x)?
f (− x ) = (− x ) ^{2} − 2 = x ^{2} − 2
Evaluating f (x) at − x does not change the function, so the function is even.

f (x) = x ^{3} + 5 x
Does evaluating f ( x ) at − x change the function? If so, is f (− x ) = −( x ^{3} + 5 x ) = − f ( x )?
f (− x) = (− x) ^{3} + 5(− x) = − x ^{3} − 5 x = −(x ^{3} + 5 x) = − f ( x )
Evaluating f (x) gives us − f (x), so the function is odd.

Does evaluating f (x) at − x change the function? If so, is − f ( x )?
Because f (− x ) is not the same as f ( x ) nor the same as − f (x), the function is neither even nor odd.
Even and Odd Functions Practice Problems
Practice
For 14, determine whether or not the graph has symmetry. If it does, determine the kind of symmetry it has. For 58, determine if the functions are even, odd, or neither.





f ( x ) = x ^{3} + 6

f ( x ) = 3 x ^{2} − 2
Solutions
 This graph has yaxis symmetry.
 This graph has x axis symmetry.
 This graph does not have symmetry.
 This graph has origin symmetry.
 f (− x ) = (− x ) ^{3} + 6 = − x ^{3} + 6

f (− x ) ≠ f (− x ) and f (− x ) ≠ − f ( x ), making f ( x ) neither even nor odd.
f (− x ) = 3(− x ) ^{2} − 2 = 3 x ^{2} − 2
f (− x ) = f (x), making f (x) even.
Find practice problems and solutions for these concepts at Functions and Their Graphs Practice Test.
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