Question

Show that
o(x)=f(x)−f(−x) is an odd function.

Answer 1

To demonstrate that o(x) = f(x) - f(-x) is an odd function, we verify that o(-x) = -o(x), which holds true, confirming o(x) is odd and integral over all space is zero.

Step-by-step explanation:

To show that the function o(x) = f(x) - f(-x) is an odd function, we need to verify that it satisfies the property of odd functions, which is o(-x) = -o(x). Let's apply this property to our function:

1. Substitute -x into o(x): o(-x) = f(-x) - f(--x) = f(-x) - f(x).
2. By multiplying the whole function by -1, we get: -o(x) = -(f(x) - f(-x)) = -f(x) + f(-x).
3. Comparing o(-x) and -o(x), we see they are equal: o(-x) = -o(x), which confirms o(x) is in fact an odd function.

This aligns with the characteristic of odd functions where the product of an odd function (like x) and an even function results in an odd function. Furthermore, this also explains why the integral of an odd function over all space is zero, as the positive and negative areas on either side of the y-axis perfectly cancel each other out.