Question

If a function f(x) is odd about a point, say (a,0), on the x-axis what exactly does this mean? How would you relate f(x values to left of a) to f(x values to right of a)?

Similarly, if a function f(x) is even about a point, say (a,0), on the x-axis what exactly does this mean? How would you relate f(x values to left of a) to f(x values to right of a)?

I understand what is meant by odd and even functions about the origin, I just want to make sure I understand how a function can be odd or even about a point on the x axis. Can you also have odd and even functions about points on the y-axis? How would you express these?

Answer 1

If a function is odd then it is symmetrical about origin (0,0) and if it is even then it is symmetrical about the y-axis, i.e., the line x=0.

Similarly, if function f(x) is odd about a point (a,0) then it means it is symmetrical about the point (a,0), in particular

f(a-x)=-f(a+x).

For an example, Sin x is symmetrical about (0,0), i.e. Sin(-x)=-Sin(x) and hence it is odd function about (0,0). Now, consider

f(x)=Sin(x-2) then f(2-x)=Sin(-x)=-Sin(x)=-f(2+x), i.e Sin(x-2) is an odd function about the point (2,0).

In general, f(x) is an odd function around (0,0), then f(x-a) is an odd function about the point (a,0).

Now, a function is even about a point (a,0) then it is symmetrical about the line x=a, i.e.

f(a-x)=f(a+x).

For example Cos(x-a) is symmetrical about the line x=a and hence even function about the point (a,0).

Let us consider a point (0,a) on the y-axis. Then a function f(x) is odd about the point (0,a) if the function g(x)=-a+f(x) is odd about the point (0,0). Moreover, a function f(x) is even about the point (0,a) if the function g(x)=-a+f(x) is even about the point (0,0).