What Is an Even Function?

In general, if any x in the domain of function f (x) has f (x) = f (-x), then the function f (x) is called an Even Function .

The earliest definition of parity function

1727, ^[1]

1.If you know the function expression, for the function f (x)

Algebraic Judgment of Even Functions

Mainly according to the definition of the parity function, first determine whether the domain is symmetrical about the origin . If it is asymmetric, it is non-odd and non-even . If it is symmetric, f (-x) =-f (x) is an odd function; f (- x) = f (x) is an even function ^[2] .

Geometric judgment of even function

The function with respect to the origin symmetry is an odd function, and the function with respect to the Y-axis symmetry is an even function.

If f (x) is an even function, then f (x + a) = f [-(x + a)]

But if f (x + a) is an even function, then f (x + a) = f (-x + a)

Even function algorithm

(1) The sum of two even functions is the even function ^[3] .

(2) The sum of two odd functions is an odd function.

(3) The sum of an even function and an odd function is a non- odd function and a non-even function.

(4) The product of the multiplication of two even functions is an even function.

(5) The product of the multiplication of two odd functions is an even function .

(6) The product of an even function and an odd function is an odd function .

(7) The odd function must satisfy f (0) = 0 (because the expression F (0) means that 0 is in the domain range, F (0) must be 0), so it is not necessary that the odd function has f (0), However , F (0) must be equal to 0 when there is F (0) , there may not be f (0) = 0, and the odd function is introduced. At this time, the function is not necessarily an odd function. For example, f (x) = x ^ 2.

(8) The odd function f (x) defined on R must satisfy f (0) = 0; because the domain is on R, f (0) exists at x = 0. To be symmetrical about the origin, at the origin It can only take one y value, which can only be f (0) = 0. This is a straightforward conclusion: when x can take 0 and f (x) is an odd function, f (0) = 0).

(9) f (x) is both an odd and an even function if and only if f (x) = 0 (the domain is symmetrical about the origin).

(10) On a symmetric interval, the definite integral whose integrand is an odd function is zero.