Whether arctanx is odd or even

Arctanx.

Bit (- Symmetry with respect to the origin.

f(x)=arctanx

The f(-x)=arctan(-x)=-arctanx=-f(x) function is an odd function.

The odd function means that for any x in the definition domain of the function f(x) with respect to the origin symmetry of a domain, there is f(-x) = - f(x), then the function f(x) is called an odd function.

Therefore, to determine whether a function is an odd function, we should first determine whether the defined domain is symmetric with respect to the origin.

f(x)=arctanx

f(-x)=arctan(-x)=-arctanx=-f(x)Therefore, the function is odd.

The basis of judging the parity of a function is to determine whether f(x) and f(-x) are equal (even functions), opposite (odd functions), or have no specific relationship (non-odd and non-even).

f(x)= arccosx。f(-x) =arccos(-x) =arccosx。

arccosx is an even function.

The above content explains:

1. In odd functions.

In f(x), the signs of f(x) and f(-x) are opposite and absolute.

equal, i.e., f(-x)=-f(x), on the contrary, the function y=f(x) of full f(x)=-f(x) must be odd. For example: f(x)=x (2n-1),n z; (f(x) is equal to x to the power of 2n-1, n is an integer) odd function.

2. The odd function image is changed to central symmetry with respect to the origin (0,0).

3. The definite delay of the odd function is annihilated.

It must be symmetrical with respect to the origin (0,0), otherwise it cannot be an odd function.

4. If f(x) is an odd function and the definition field contains 0, then f(0)=0.

Non-odd and non-even functions. Upstairs when fart, I didn't even think about defining the domain.

y=arcsinx, and the domain is defined as [-1,1].

arcsin(-x)=-arcsinx, y=arcsinx is an odd function.

Its image is below.

arctanx is an odd function. f(x)=arctanx f(-x)=arctan(-x)=-arctanx=-f(x), the basis of judging the parity of a function is to determine whether f(x) and f(-x) are equal (even function), opposite (odd function), or have no specific relationship (non-odd and non-even).

1. The difference between the sum or subtraction of two odd functions is the odd function.

2. The product obtained by multiplying two odd functions or the quotient obtained by clear balance and division is an even function.

3. The product of an even function multiplied by an odd function or the quotient obtained by division is an odd function.

4. The difference between the sum or subtraction of an even function and an odd function is a non-odd and non-even function.

5. F(x) is both an odd and even function if and only if f(x)=0 (the domain is wide symmetric with respect to the origin). The integral of the odd function on the symmetry interval is zero.

1) Odd functions have the same monotonicity within the symmetrical monotonic interval.

Even functions have opposite monotonicity within a symmetrical monotonic interval.

2) If f(x-a) is an odd function, then the image of f(x) is symmetrical with respect to points (a,0).

If f(x-a) is an even function, then the image of f(x) is symmetrical with respect to the line x=a.

3) On the commonally defined domain of f(x), g(x): 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 Functions Even = Odd Functions.

arctanx is the odd envy surplus high function, which defines the domain r, and the value range [- 2, 2] brother ruler.

tanx0=-1

What about arctan-1=x0

But note that the tanx period is , x0=- 4+n and arctan-1=- 4 (because of the range [- 2, 2]) <

Arctanx image.

tanx image.