Even and odd functions:
If f: X→Y is a original valued function such that for all x∈D => - x ∈ D (where D = domain of f) and if f(- x) = f(x) for every x∈D then f is called an even function and if f(- x) = - f(x) then f is called be odd function.
MORE IMPORTANT POINTS:
- Even functions are symmetric with related to the y - axis (i.e. if (x, y) lies on the curve, then (- x, y) also lies on the curve).
- Odd functions are symmetric about the origin and it is located either in the first and third quadrant or in the second and fourth quadrant. (i.e. if (x, y) place in between on the graph, then (- x, - y) also lies on the curve).
- f(x) = 0 is the only function which is both odd and even.
- If f(x) is an odd function, then f(x) is an odd function shown f(x) is differentiable on R.
- To express a provided function f(x) as the summation of an odd and even function, we write
is an even function and
is an odd function.
- If x = 0 ∈ domain of f, then for odd function f(x), f(0) = 0 i.e. but for a function, f(0) ≠ 0, then that function may not be odd.
Problem: Which of the subsequent functions is (are) odd, even or neither :
![2167_Even and odd functions2.png](https://www.expertsmind.com/../CMSImages/2167_Even and odd functions2.png)
Solution:
![692_Even and odd functions3.png](https://www.expertsmind.com/CMSImages/692_Even%20and%20odd%20functions3.png)
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