Given the function f (x) = (12x-1 + 12) SiNx & nbsp; (- π 2 < x < π 2 and X ≠ 0) (1) judge the parity of F (x); (2) prove that f (x) > 0

Given the function f (x) = (12x-1 + 12) SiNx & nbsp; (- π 2 < x < π 2 and X ≠ 0) (1) judge the parity of F (x); (2) prove that f (x) > 0

(1) ∵ f (- x) = (12-x-1 + 12) sin (- x) = - (112x-1 + 12) SiN x = - (2x1-2x + 12) SiN x = (2x2x-1-12) SiN x = [(1 + 12x-1) - 12] SiN x = (12x-1 + 12) SiN x = f (x), f (x) is even function. (2) when 0 ＜ x ＜ π 2, 2x ＞ 1, & nbsp; & nbsp; 2x-1 ＞ 0 & nbsp; & nbsp
Constant function, inverse proportion function, positive proportion function, first-order function, second-order function, check function, sibling function, inverse proportion function, absolute value function, exponential function, logarithmic function, power function, etc. What are the domain of definition, image, range of value, monotone interval and parity?
1. Definition domain of constant function: R image: parallel to X axis and the distance to X axis is a straight line of the constant. Value domain: {value of the constant} monotone interval: (- ∞, + ∞) does not increase or decrease parity: even function, symmetric about y axis; 2. Definition domain of inverse scale function: all real number images of X ≠ 0 & amp; # 61625;: value domain: y ≠ 0 & amp; # 61625; All real monotone intervals: Parity: odd function, image centrosymmetry about origin 3. Definition domain of positive proportion function: R value domain: R image: monotone interval: Parity: odd function, image centrosymmetry about origin 4. Definition domain of primary function: R image: value domain: R monotone interval: K & gt; 0, increasing, K & lt; 0, In general, the relationship between the independent variable x and the dependent variable y is as follows: y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0, and a determines the opening direction of the function. When a & gt; 0, the opening direction is upward, when a & lt; 0, the opening direction is downward, and IAI can also determine the opening size. The larger IAI is, the smaller the opening is, and the smaller IAI is, the larger the opening is.) The right side of quadratic function expression is usually quadratic trinomial. II. Three kinds of expressions of quadratic function: y = ax ^ 2; + BX + C (a, B, C are constants, a ≠ 0) vertex expression: y = a (X-H) ^ 2; + K [vertex P (H, K)] intersection expression: y = a (x-x1) (x-x2) [only a (x1,0) and B (x2,0) with X axis, Note: in the mutual transformation of the three forms, there are the following relations: H = - B / 2a, k = (4ac-b ^ 2;) / 4A, x1, X2 = (- B ± √ B ^ 2; - 4ac) / 2a, III, The image of quadratic function is a parabola. IV. properties of parabola 1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line x = - B / 2A. The only intersection point between the axis of symmetry and the parabola is the vertex P of the parabola. In particular, when B = 0, the axis of symmetry of the parabola is the Y axis (that is, the straight line x = 0). 2. The parabola has a vertex P, and the coordinate is p [- B / 2a, (4ac-b ^ 2;) / 4A]. When - B / 2A = 0, the parabola has a vertex P, When a ＞ 0, the parabola opens upward; when a ＜ 0, the parabola opens downward. The larger | a | is, the smaller the parabola's opening is. 4. The first-order coefficient B and the second-order coefficient a jointly determine the position of the axis of symmetry, 5. The constant term C determines the intersection of the parabola and the y-axis. The intersection of the parabola and the y-axis lies in (0, c). 6. When the number of intersections of the parabola and the x-axis is Δ = B ^ 2-4ac ＞ 0, the parabola and the x-axis have two intersections. When Δ = B ^ 2-4ac = 0, the projectile line and the x-axis have one intersection, There is no intersection point between parabola and X axis. V. quadratic function and quadratic equation of one variable. In particular, quadratic function (hereinafter referred to as function) y = ax ^ 2; + BX + C, when y = 0, quadratic function is quadratic equation of one variable (hereinafter referred to as equation) about X, namely ax ^ 2; + BX + C = 0, The abscissa of the intersection of function and X axis is the root of the equation. When drawing a parabola y = AX2, you should first list, then trace the point, and finally connect the line. When selecting the independent variable x value in the list, always take 0 as the center, select the integer value that is easy to calculate and trace the point. When tracing the line, you must use a smooth curve, There are several forms of quadratic function analytic formula: (1) general formula: y = AX2 + BX + C (a, B, C are constants, a ≠ 0); (2) vertex formula: y = a (X-H) 2 + K (a, h, K are constants, a ≠ 0); (3) two formulas: y = a (x-x1) (x-x2), where X1 and X2 are abscissa of intersection point of parabola and x-axis, that is, two roots of quadratic equation AX2 + BX + C = 0, A ≠ 0. Note: (1) any quadratic function can be transformed into vertex formula y = a (X-H) 2 + K by formula, the vertex coordinate of parabola is (h, K), when h = 0, the vertex of parabola y = AX2 + k is on the y-axis; when k = 0, the vertex of parabola a (X-H) 2 is on the x-axis; when h = 0 and K = 0, the vertex of parabola y = AX2 is on the origin, Let y = ax ^ 2; if the axis of symmetry is Y axis, but not the origin, then let y = ax ^ 2 + K define and define the expression. Generally, there is the following relationship between the independent variable x and the dependent variable y: y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0, and a determines the opening direction of the function. When a & gt; 0, the opening direction is up, and when a & lt; 0, the opening direction is down. IAI can also determine the opening size, and the larger IAI, the smaller the opening, The smaller IAI is, the larger the opening is.) then y is called the quadratic function of X. the right side of the quadratic function expression is usually the quadratic trinomial. X is the independent variable, and Y is the function of X. there are three kinds of expressions of quadratic function: (1) general formula: y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0). (2) vertex formula [vertex P (h, K)]: y = a (X-H) ^ 2 + K, 0): y = a (x-x1) (x-x2) the above three forms can be transformed as follows: ① the relationship between the general formula and the vertex formula. For the quadratic function y = ax ^ 2 + BX + C, the vertex coordinates are (- B / 2a, (4ac-b ^ 2) / 4a), that is, H = - B / 2A = (x1 + x2) / 2K = (4ac-b ^ 2) / 4A. ② the relationship between the general formula and the intersection formula x1, X2 = [- B ± √ (b ^ 2-4ac)] / 2A (i.e. the root formula of quadratic equation of one variable) 6. Definition domain of check function: Image: value domain: monotone interval: Parity: 7. Definition domain of sibling function: Image: value domain: monotone interval: Parity: 8. Class inverse proportion function: F (x) = K / (x + C) + B. the problem can be solved by translation 9. Definition domain of absolute value function: all real value ranges: all Non negative image: Parity: absolute value function is even function, and its graph about Y-axis symmetric monotone interval: (- ∞, 0) decreasing, (0, + ∞) increasing 10. Definition domain of exponential function: Image: value domain: monotone interval: Parity: 11. Definition domain of logarithmic function: image: value domain: monotone interval: Parity: 12. Definition domain of power function: Image: value domain: monotone interval: Parity: 12
F (x) = the fourth power of X + X to judge the parity,
f(x)=x^4+x
f(-x)=(-x)^4+(-x)=x^4-x
So: F (- x) is not equal to f (x), or - f (x)
So: F (x) is not odd or even
On linear function, inverse proportion function and positive proportion function
(1) In the same rectangular coordinate system, the first-order function and the inverse proportion function intersect at two points, one of which is (a, b), then the other is (- B,
(2) In the same rectangular coordinate system, the positive scale function and the inverse scale function intersect at two points, one of which is (a, b), then the other is (- A, b),
(1) In the same rectangular coordinate system, the first-order function and the inverse proportion function intersect at two points, one of which is (a, b), then the other is (- B, - a)?
not always.
(2) In the same rectangular coordinate system, the positive scale function and the inverse scale function intersect at two points, one of which is (a, b), then the other is (- A, - b)?
Sure. Because both positive and inverse scale function images are symmetrical about the origin
(1) In the same rectangular coordinate system, the first-order function and the inverse proportion function intersect at two points, one of which is (a, b), then the other is not necessarily (- B, - a)
(2) In the same rectangular coordinate system, the positive scale function and the inverse scale function intersect at two points. If one point is (a, b), then the other point must be (- A, - b)
If you have any questions, please ask; if you are satisfied, please accept, thank you! ... unfold
(1) In the same rectangular coordinate system, the first-order function and the inverse proportion function intersect at two points, one of which is (a, b), then the other is not necessarily (- B, - a)
(2) In the same rectangular coordinate system, the positive scale function and the inverse scale function intersect at two points. If one point is (a, b), then the other point must be (- A, - b)
If you have any questions, please ask; if you are satisfied, please accept, thank you! Put it away
Judging the parity of the third power SiNx of F (x) = cos (2 π - x) - x
The third power SiNx of F (x) = cos (2 π - x) - x
The third power sin (- x) of F (- x) = cos (2 π + x) + X
=The third power of COS (2 π - x) - x SiNx = f (x)
Even function
The relationship between the intersection coordinates of positive scale function and inverse scale function
Because: the image of positive scale function is centrosymmetric about the origin
The image of inverse scale function
So: the intersection of positive and negative scale functions is also centrosymmetric about the origin!
The intersection of positive and negative scale functions is also centrosymmetric about the origin!
Because: the image of positive scale function is centrosymmetric about the origin
The image of inverse scale function
Then the intersection is also centrosymmetric about the origin! What about linear function and inverse proportion function? A function of degree, such as y = KX + B, the intersection point should be symmetric about a point on the line. Specifically, you can use the same distance to solve. It's not hard. ... unfold
The intersection of positive and negative scale functions is also centrosymmetric about the origin!
Because: the image of positive scale function is centrosymmetric about the origin
The image of inverse scale function
Then the intersection is also centrosymmetric about the origin! What about linear function and inverse proportion function?
Make an image of the function y = | 3 ^ X-1 | and point out the value of K. when the equation | 3 ^ X-1 | = k has no solution, how to draw the image?
Image
It is known from the graph that when: K & lt; 0 & nbsp; there is no solution;
&There is a solution when k = 0;
&There are two solutions when there are & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; K & gt; 0 & nbsp
Positive proportion function and inverse proportion function in junior high school mathematics
1、 Why can't the positive scale function K be 0? Why must the degree of X be 1?
2、 Why can't the inverse proportion function K be 0? When the form of the inverse proportion function is y = KX (- 1), why must the degree of X be - 1 instead of other negative numbers? In other words, why must the degree of X be 1 when y = K / x?
If K is zero, it becomes a constant function, that is y = 0 * x + B, that is y = B (constant). On the image, it is a horizontal line, which is not proportional to the independent variable. The number of times must be one, because if it is higher than one, it is not a straight line, it is not a parabola, so it is not proportional (proportional is a rising straight line)
Inverse proportion function, if k = 0, then y = 0. This function is the x-axis, which is not inversely proportional to the independent variable. Similarly, if the degree is not negative one, then it is not an inverse proportion function that works with the increase of the value of the independent variable
Draw the graph of the function y = | 3x-1 | and use the graph to answer: when k is the value, the equation | 3x-1 | = k has no solution? Is there a solution? There are two solutions?
The image of y = | 3x-1 | is as follows: when k < 0, | 3x-1 | = k has no solution; when k = 0 or k > 1, | 3x-1 | = k has one solution; when 0 < K < 1, the equation | 3x-1 | = k has two solutions
The definition of function in junior high school
In a certain process of change, there are two variables X and Y. if x is given a value, y will have a unique and definite value, then x is an independent variable and Y is called a function of X,
It's a model to study the relationship between variables, such as when you buy instant noodles,