Increasing Functions Subtracting Functions Subtracting Functions, What is an Increasing Function Wha

Only the addition of functions with the same increase or decrease can be judged by the increase or decrease of the function, if not, it should be judged by the image of the "Nike" function.

For example, the function y=x+(1 x), when x=1 x, the two x values are the abscissa of the inflection point of the Nike function image in the figure below, and the increase or decrease of the function is judged according to the image.

However, it should be noted that the two x values must be opposite to each other, so not any two functions that increase and decrease add up to be a "Nike" function.

When the monotonicity is different, the two functions are added together, and its monotonicity cannot be judged.

This definition of the composite function is different, and the monotonicity of the definition of the composite function conforms to the principle of the same increase and difference decrease. For example, if f(x) is an increasing function, then f(x+1) is also an increasing function and f(-x+2) is a decreasing function.

Increment functionThat is, as x increases, y increases, e.g. y=x.

Subtract the functionThat is, as x increases, y decreases, e.g. y=1 x.

Functions, first developed by the Chinese Qing Dynasty mathematician Li Shanlan.

Translation, from his book Algebra

The reason for this translation is that "where there is a variable in this variable, then this is a function of the other", that is, a function refers to the change of a quantity with the change of another quantity, or that one quantity contains another quantity. <>

Related Concepts: In a process of change, the amount of change is called a variable (in mathematics, the variable is x, and y changes with the value of x), and some values do not change with the variable, we call them constants.

Argument. Function): A variable associated with a quantity in which any value can find a fixed value.

Dependent variable. Function): Varies with the change of the independent variable, and when the independent variable takes a unique value, the dependent variable (function) has and only a unique value corresponding to it.

Function value: In a function where y is x, x determines a value, y determines a value, and when x takes a, y is determined as b, and b is called the function value of a.

When x1>x2

f(x) is an increment function.

f(x1) > f(x2) can be derived

g(x) is a subtractive function.

g(x1) < g(x2) can be derived

The relationship between the increasing and decreasing functions is as follows:

Increasing function + increasing function = increasing function, increasing function - subtracting function = increasing function, decreasing function + decreasing function = decreasing function, decreasing function - increasing function = decreasing function. The increase or decrease of the increase function + subtraction function is not necessarily.

In general, let the function f(x) be defined in the domain d, if for an interval within the defined domain d.

The value of any two independent variables is x1, x2, when x1

Proof:

Odd functions f(-x) = -f(x), g(-x) = -g(x).

Even function h(-x) = h(x).

i（x）=f（x）+g（x）

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

j（x）=f（x）-g（x）

j（-x）=f（-x）-g（-x）=-f（x）-（g（x））=f（x）-g（x）=-j（x）

Odd functions are added, and odd functions are subtracted to become odd functions.

Additive-even function, subtraction-even function, not necessarily.

The relationship between the addition and subtraction of the increase function and the subtraction function is also not necessarily.

An increasing function is a monotonically increasing interval and a decreasing function is a monotonically decreasing function over a specified interval, provided that the function is continuous.

Increment function subtract increment function : That is not certain.

Example 1h(x) =x

g(x) =2x

h(x) and g(x) are both increments.

Cause. f(x) = h(x) -g(x): increase the function and subtract the increase function.

x -2xx is a subtractive function.

Example 2h(x) =2x

g(x) =x

h(x) and g(x) are both increments.

Cause. f(x) = h(x) -g(x): increase the function and subtract the increase function.

2x -xx is an increment function.

Example 3h(x) =x

g(x) =x^3

h(x) and g(x) are both increments.

Cause. f(x) = h(x) -g(x): increase the function and subtract the increase function.

x -x^3

This is not an additive function or a subtraction function.

A subtractive function is a function in which the value of a function decreases with the increase of the independent variable and increases with the decrease of the independent variable within the defined domain. For example: y=-x; y=1 to the xth power of 2, etc.

In mathematical terms, for the function y=f(x) that defines the domain d, if any x1,x2 satisfies x1,x2 d, and x1 >x2, then there is f(x1) f(x2).

1) Increment function + Increment function = Increment function;

2) Subtractive function + subtractive function = subtractive function;

3) Increasing function - subtracting function = increasing function;

4) Subtraction function - increase function = subtraction function.

The domain of the function f(x) is i, and if for the values x1 and x2 of any two independent variables on an interval d in the defined domain i, when x1f(x2), then f(x) is said to be a decreasing function in this interval, and the interval d is called a decreasing interval. The image of the subtractive function is descending from left to right, i.e., the value of the function decreases as the independent variable increases. Whether a function is subtractive can be determined by definition, image, intuition, or by using the positive and negative derivatives of the interval.

The increase function subtraction function is an increase function or meaningless, e.g. y1 = (x 2-4) (x>=2), and y2 = (1-x 2) (0<=x<=1), y1-y2 is meaningless.

The increase function is the decrease functionThe definition of the function is given a set of numbers a, and the corresponding law f is applied to a, which is denoted as fa, and another set of numbers b is obtained, that is, b is equal to fa, then this relation is called a function relation, referred to as a function, and the concept of function contains three elements, defining the domain a, the value range c and the corresponding law f, the core of which is the corresponding law f, which is the essential feature of the function relationship.

Functions were first translated by Li Shanlan, a mathematician of the Qing Dynasty in China, because of the reason why his book Algebra translated it this way, the reason he gave was that if there is a variable in this variable, then this is the function of the other, that is, the function refers to the change of a quantity with the change of another quantity, or that a quantity contains another quantity.

Introduction to functions. The word function used in Chinese mathematics books is a translation of the word Li Shanlan, a mathematician of the Qing Dynasty in China, translated function into function when translating the book "Algebra" in 1859.

The definition given by Li Shanlan is that the formula contains heaven in the ordinary formula, which is a function of heaven, and ancient China used 4 words of heaven and earth characters to represent 4 different unknowns or variables.

The implication of this definition is that if the formula contains the variable x, then the formula is called the function of x, so the function refers to the meaning that the formula contains variables.

The increment function is to increase with x and y, e.g. y=x

The subtraction function decreases as x increases, e.g. y=1 x

The expression of the primary function is y=kx+b, x can take any real number, as long as k0, the primary function is an increasing function.

Extended Materials. A method for judging monotonicity.

1) Definition method: that is, "take the value (within the definition domain) to make the difference, deform, fix the number, and judge";

2) Image method: first make a function image, and use the image to intuitively judge the monotonicity of the function;

3) Direct method: It is to write the monotonic interval of the functions we are familiar with, such as the primary function, the quadratic function, the inverse proportional function, etc.

4) Derivative: Suppose that the function f is continuous over the interval [a,b] and differentiable on (a,b), if each point x (a,b) has f'(x) >0, then f is incrementing on [a,b]; If each point x (a, b) has f'(x) <0, then f is decreasing on [a,b].