Determine the parity of the function y sin 3 4x 3 2x

f(x)=y=sin(3x/4+3π/2)sin(3x/4+3π/2-2π)

sin(3x/4-π/2)

sin(π/2-3x/4)

cos(3x/4)

f(-x)=-cos(-3x 4)=-cos(3x 4)=f(x) defines the domain as r

So it's an even function.

First of all, the landlord's question input format confuses me, I can understand it in 2 ways: 1]y=sin((3 4)x+(3 2)x)2]y=sin((3 4x)+(3 2x)), either of which can be proved with the following steps.

Solution: 1Obviously, the original function defines the domain with respect to the origin symmetry.

2.∵f(-x)=sin(-(3/4)x-(3/2)x)=-sin((3/4)x+(3/2)x)=-f(x)

or f(-x)=sin(-(3 4x)+(3 2x))=-sin((3 4x)+(3 2x))=-f(x).

That is, f(-x)=-f(x), so the original function is an odd function ps: The above solution is actually a general step to judge the parity of the function, and I wish the landlord to master the knowledge of the function as soon as possible.

Answer: y=sin(2x+3 vulture 2).

y=sin(2x+2-1 2*vulture).

y=sin(2x-vulture 2).

y=-sin(煀 2-2x).

y=-cos(2x)

y=-(cosx)^2+(sinx)^2

So f(-x)=-(cos-x) 2+(sin-x) 2=-(cosx) 2+(sinx) 2=f(x).

So the original function is an even function.

f(x)=-f(-x) is the odd function.

f(x)=f(-x) is an even function.

If you haven't learned an arbitrary function yourself, use this method.

1. y=sinx

1. Parity: odd functions.

2. Image nature:

Central symmetry: Symmetry with respect to the point (k,0).

Axisymmetry: Symmetry with respect to x=k + 2.

3. Monotonicity:

Increase interval: x [2k - 2, 2k + 2] subtract interval: x [2k + 2, 2k + 3 2] 2, y = cosx

1. Parity: even function.

2. Image nature:

Central symmetry: Symmetry with respect to the point (k + 2,0).

Axisymmetry: Symmetry with respect to x=k.

3. Monotonicity:

Increment range: x [2k - 2k].

Minus interval: x [2k, 2k+

3. y=tanx

1. Parity: odd functions.

2. Image nature:

Symmetry of the center of the sail bucket: symmetry with respect to the point (k 2,0).

3. Monotonicity:

Increase interval: x (k - 2, k + 2).

There is no reduction interval.

Four-state slow, y=cotx

1. Parity: odd functions.

2. Image nature:

Central symmetry: Pin is symmetrical with respect to the point (k 2,0).

3. Monotonicity:

Subtraction: x (k, k +

There is no increase in the interval.

It's not strange, it's boring, it's not even.

First of all, the definition of the communication domain is about the hunger bend of origin symmetry.

and f(-x)=2 (sin(-x))=2 (-sinx)≠-2 (sinx)≠2 sinx

Obviously, it does not satisfy both odd and even function conditions, so it is a non-odd and non-even function.

Define the domain as r, with respect to the origin symmetry On r, there is a clear hail: f(-x)+f(x)=-x+sin(-x)+x+sin(x) =x-sin(x)+x+sin(x)=0 Therefore, the function y=x+sinx is the number of odd sails on r.

Judgment: Because y(-x)=-sinx-cosx+1≠y(x), it is not equal to -y(x), so it is not odd or even.

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 with respect to the symmetry of the origin, then the function must not be an odd (or even) function.

To judge the parity of a function, first of all, to test whether the definition domain is symmetrical at the origin, and then to draw conclusions by simplifying, sorting, and then comparing it with f(x) in strict accordance with the definition of odd and evenness), and the basis for judging or proving whether the function has parity is the definition and variant.

The function y=3x+2 is an odd function.

A function f(x) is an odd function if and only if for all x, there is f(-x)=-f(x). In this case, the image of this function is symmetrical in the origin of the forest roll.

For the number of this hall y=3x+2, there is:

f(-x) =3(-x) +2 = 3x + 2f(x) =3x + 2

Since f(-x)=-3x+2 and f(x)=3x+2, so f(-x)=-f(x), so y=3x+2 is an odd function.

Because f(-x) ≠ f(x), it is neither an odd nor an even function.

Assumption: f(x)=y x r f(-x)=sin(3 2 -x)=cos(-x)=cos(x) Description: Zaopeisin(3 2 +x) is equivalent to the right translation of the x-axis 3 2, or to the left of the cherry blossom collapse 1 2 , so it becomes cos(x) f(x)=sin(3 2 +x)=cos(x) and the whole function y is continuous, y=sin(3 2 +x) is an even function.

Solution: Let f(x)=y=x +sinx

x takes any real number, the function expression is always meaningful, the function definition domain is r, and the origin is symmetrical.

f(-x)=(-x) +sin(-x)=x -sinxf(-x)+f(x)=x -sinx+x +sinx=2x, which is not constant 0, and the function is not an odd function.

f(-x)-f(x)=x -sinx-x -sinx=-2sinx, which is not constant to 0, and the function is not even.

Functions are non-odd and non-even.