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Concept

Odd Function

An odd function is a function for which the value of f(-x) is equal to the value of -f(x) for all the values in its domain. It is like if the function allows moving the negative sign from the input to the output.


f(- x) = -f(x)

The graph of an odd function is symmetric about the origin, meaning that the graph looks the same after a 180^(∘) rotation about the origin. The functions y=x and y=x^3 are two examples of odd functions.
Graphs of f(x)=x and graph of g(x)=x^3
Notice that if a function is odd and the point (x,y) is on the graph, then the point (-x,-y) is also on the graph. A function can be odd, even, or neither.

Extra

 Determining If a Function is Odd
To determine algebraically whether a function is odd, substitute -x into the function rule and simplify. If the resulting expression is equal to -f(x), then the function is odd; otherwise, it is not. For example, consider the following function. f(x) = x/x^2-1 Substitute -x for x and simplify.
f(x) = x/x^2-1
Substitute

x= -x

f( -x) = -x/( -x)^2-1
NegBaseToPosBase

(- a)^2=a^2

f(-x) = -x/x^2-1
MoveNegNumToFrac

Put minus sign in front of fraction

f(-x) = -x/x^2-1
Substitute

x/x^2-1= f(x)

f(-x) = - f(x)
Since f(-x)=-f(x), the given function is odd.
Exercises

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