Function parity determination, how to determine function parity

Parity: odd function + odd function = odd function.

Even function + even function = even function.

Odd function * odd function = even function.

Even function + even function = even function.

Odd function * even function = odd function.

It means: the parity of a new function obtained by the sum and product of an odd or even function and another odd or even function!!

For example: "function + odd function = odd function" means that the function composed of the sum of one odd function and another odd function is still an odd function! The same goes for everything else!

In monotonicity: increase + increase = increase.

minus + minus = minus.

Increase-decrease = increase.

Decrease - increase = decrease.

The meaning of the representation is: the monotonicity of the sum of one monotonic function and the product of another monotonic function and the product of the new function!!

For example: "increase + increase = increase" means that the function composed of the sum of one monotonic increase function and another monotonic increase function is still a monotonic increase function! The same goes for everything else!

Look. Suppose f(x) and g(x) are even functions, and h(x) and j(x) are odd functions.

odd function + odd function = odd function, proof: s(x) = h(x) + j(x), s(-x) = h(-x) + j(-x) = -h(x)-j(x) = -s(x).

The rest you can prove in this way.

Both increasing functions increase sequentially, can he add two without increasing it?

The two subtraction functions are decreasing in turn, can he add the two without subtracting?

Increase-decrease = increase, prove: increase-decrease = increase + (-decrease), i.e., -decrease is the increase function, so increase-decrease = increase.

Decrease-increase = decrease, prove: decrease-increase = decrease + (-increase), that is, - increase is the reduction function, so it can be proven.

The odd function has f(-x)=-f(x), so the odd function + odd function = odd function can be proved as follows:

If f(x) and g(x) are odd functions, h(x) = f(x) + g(x).

h(-x)=f(-x)+g(-x)=-f(x)-g(x)=-[f(x)+g(x)]=-h(x)

So h(x) is an odd function.

The even function has f(-x)=f(x), so the even function + even function = even function can be proved as follows:

If f(x) and g(x) are even functions, h(x) = f(x) + g(x).

h(-x)=f(-x)+g(-x)=f(x)+g(x)=h(x)

So h(x) is an even function.

This can be proved in this way by all other equations.

If it is a judgment, the odd function proves f(-x)=-f(x), and the even function proves f(-x)=f(x).

If the odd function proves f(-x)=-f(x) and the even function proves f(-x)=f(x) is not easy to analyze, then the addition and subtraction relationship between the two functions can be used to prove the indirect function, and the indirect proof can be used such as odd function + odd function = odd function.

The same is true for the increase and decrease function.

Increment function: if xf(y)

Use the definitions of odd and even functions to judge:

1) Whether the definition domain is symmetrical or not, the definition domain of odd or even functions is symmetrical, if the definition domain is asymmetric, it is neither an odd function nor an even function;

2) The odd function satisfies f(-x) = -f(x).

3) The even function satisfies f(-x)=f(x).

The mantra for judging the parity of functions is known as follows: the inner odd is even, and the inner odd is the same as the outside. Prerequisites for verifying parity: The definition domain of the function must be symmetric with respect to the origin.

1. First, decompose the function into a general function that is always seen in the chain, such as polynomial x n, trigonometric function, and judge parity.

2. Judging from the operation rules between the decomposed functions, there are generally only three kinds of f(x)g(x), f(x)+g(x), and f(g(x)) (division or subtraction can become the corresponding multiplication and addition).

3. If one of f(x) and g(x) is an odd function and the other is an even function, then f(x) g(x) odd, f(x) + g(x) are non-odd and non-even, and f(g(x)) is odd.

4. If f(x) and g(x) are even functions, then f(x)g(x)even, f(x)+g(x)even, f(g(x)) even.

5. If f(x) and g(x) are both odd functions, then f(x)g(x) even, f(x)+g(x) odd, f(g(x)) odd.

In general, for the function f(x).

If there is f(x)=f(-x) or f(x) f(-x)=1 for any x in the function f(x) definition field, then the function f(x) is called an even function. With respect to y-axis symmetry, f(-x) = f(x). For example, f(x) x 2, if for any x in the domain of the function f(x) definition, there is f(-x)=-f(x) or f(x) f(-x)=-1, then the function f(x) is called an odd function.

With respect to origin symmetry, -f(x) = f(-x). For example, f(x) x 3, if for any x in the function definition domain, there are f(x)=f(-x) and f(-x)=-f(x),(x r, and r is symmetric with respect to the origin. Then the function f(x) is both odd and even, and is called both odd and even.

If for the function definition there is a a such thing that f(a) ≠ f(-a) and a b such that f(-b) ≠-f(b), then the function f(x) is neither odd nor even, and is called a non-odd and non-even function.

The defined domains are opposites of each other, and the defining domains must be symmetrical with respect to the origin.

In particular, f(x)=0 is both an odd and an even function.

Note: Odd and evenness are integral properties of a function, for the entire defined domain.

The domain of an odd and even function must be symmetrical with respect to the origin, and if the domain of a function is not symmetrical with respect to the origin, then the function must not be parity.

Analysis: To judge the parity of a function, first test whether the defined domain is symmetrical with respect to the origin, and then simplify and sort it out in strict accordance with the definition of odd and evenness, and then compare it with f(x) to draw conclusions).

The basis for judging or proving whether a function is parity is by definition.

If an odd function f(x) is meaningful at x=0, then the function must have a value of 0 at x=0. And about the origin symmetry.

If the domain of the function definition is not symmetrical with respect to the origin or does not meet the conditions of odd and even functions, it is called a non-odd impulse grip non-even function. For example, f(x)=x [-2] or [0,+ defines the domain is not symmetric with respect to the origin).

If a function conforms to both odd and even functions, it is called both odd and even. For example, f(x)=0

Note: Any constant function (which defines the domain symmetry with respect to the origin) is even, and only f(x)=0 is both odd and even.

It is known that the function y=f(x) is defined as r, and for any a, b belongs to r, there are f(a+b)=f(a)+f(b) and when x is greater than 0.

f(x) is less than 0 of the selling ruler

Verify that f(x) is an odd function.

Proof: let a=x, b=-x

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

then f(x)+f(-x)=f(0).

Then chain Hu to take a=b=0

then f(0) = 2f(0).

then f(0)=0

So f(x)+f(-x)=0

then f(x) is an odd function.

There are techniques for visually determining the parity of certain kinds of functions without having to be proven by definition. This is helpful for multiple-choice, true/false questions.

First of all, the function that defines the symmetry of the domain to the origin can be an odd function or an even function, and the function that defines the domain is not symmetric to the origin must be a non-odd and non-even function. For example, y=x (x-1) (x-1)=x (x≠1), the definition domain is not symmetric to the origin, so it is a non-odd and non-even function.

Clause. Second, we must first be familiar with some common odd and even functions, for example, the odd power of x (including negative odd numbers such as -1 and -3) is an odd function, the even power of x (including negative even numbers such as -2 and -4) is an even function, the constant function is an even function, the even root of x is a non-odd and non-even function, the odd root of x is an odd function, the sine function is an odd function, the cosine function is an even function, the constant function is an even function, the constant function equal to 0 is both an even function and an odd function, and so on.

Clause. 3. Keep in mind some ways to infer the parity of a new function from a known function. There are several cases.

1. The new function has several functions to add and subtract, and each addition and subtraction function is an even function, then the new function is an even function, for example, x 4+x +3, x 4, x are even functions, so the new function x 4+x +3 can be directly judged to be an even function;

Each additive function is an odd function, then the new function is an odd function, for example, x 5+x 3+x, x 5, x 3, x are all odd functions, so you can directly judge that x 5+x 3+x is an odd function.

If part of the function of addition and subtraction is odd and part is even, then the new function is non-odd and non-even. For example, x + x + 4, x and 4 are even functions, and x is odd functions, so x + x+4 is non-odd and non-even functions.

2. The new function is formed by the multiplication and division of several functions, and each function of multiplication and division is an odd function or an even function (there can be no non-odd and non-even functions in the factor), then there are odd odd functions in the function of multiplication and division, and the new function is an odd function; There are even odd functions, and the new function is the odd function.

For example, xsinx, where x and sinx are both odd functions, which are multiplied by two odd functions, so xsinx is an even number; xcosx, x is an odd function, cos is an even number, there is 1 odd function, so xcosx is an odd function; x cosx, there are no odd functions, so x cosx is an even function.

3. Composite function, which is more complicated, is generally more reliable to derive by definition.

How to judge the parity of a function.

The original formula for definite integral parity.

The definition of an even function is f(x)=f(-x), so if the domain is symmetrical, just prove that f(x)=f(-x) is an even function.