Odd functions are those that satisfy the condition f(-x)=-f(x) For example, check if x^3 is odd => f(x)=x^3 f(-x) = (-x)^3 -f(x)=-x^3 Since (-x)^3=-x^3, we see that f(x)=x^3 is an odd function. In fact, polynomials which contain odd-powered terms only are odd. (constant is even) As an exercise, you can verify that sin(x) is odd, cos(x) is even.
On graphs, odd functions are those that resemble a 180 degree rotation. Check with graphs of above examples.
So we identify the first graph (f(x)=-x^3) is odd (we can identify a 180 degree rotation)
Odd functions have a property that the sum of individually odd functions is also odd. For example, x+x^3-6x^5 is odd, so is x+sin(x).
For the next graph, f(x)=|x+2| is not odd (nor even) because if we rotate one part of the graph, it does not coincide with another part of the graph, so it is not odd.
For the last graph, f(x)=3cos(x), it is not odd, again because if we rotate about the origin by 180 degrees, we get a different graph. However, it is an even function because it is symmetrical about the y-axis.
