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MindMap Gallery Function in the first grade of high school

Function in the first grade of high school

[The Beauty of Mathematics: The Mystery of Unlocking Functions] Functions are the magical link between variables. In the first grade of high school, "Functions" will take you to explore core concepts such as basic elementary functions and composite functions, from explicit y=f(x to implicit function F(x,y)=0, from inverse function to intermediate variables of composite functions, revealing the four major properties of variable relationships (parity, periodicity, etc.) layer by layer, just like function DNA, the definition domain and value domain define its boundaries, especially the hyperbolic function, Dirichlet function and other typical examples, showing the functional universe from rules to change.

Edited at 2025-09-06 08:35:54

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Function in the first grade of high school

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Outline

"Function" in the first grade of high school

Common functions

Absolute value function

y=|x|=

Symbol Functions

y=sgn x=

Rounding function

y=[x]

Dirichlet function

y=D(x)=

Most value function

y=max{f(x),g(x)}, y=min{f(x),g(x)}

Variable integral upper limit function

Suppose the function f(x) is continuous in the interval [a,b], and x∈[a,b], if the upper limit x of the range [a,b] changes arbitrarily, for each fixed x value, the fixed integral has a corresponding value, so it defines a function on [a,b]. Called w the upper limit function of variable integral

Hyperbolic Function

Hyperbolic sinusoidal function

y=sh x=

Hyperbolic cosine function

y=ch x=

Hyperbolic Tangent Function

y= th x=

Hyperbolic co-tangent function

y=cth x=

Hyperbolic positive cutting function

y=sech x=

Hyperbolic resection function

y=csch x=

Inverse hyperbolic sinusoidal function

y=arsh x=

Inverse hyperbolic cosine function

y=arch x=

Inverse hyperbolic tangent function

y=arth x=

Basic elementary functions and elementary functions

Basic elementary functions

Power function

y=: y=x², y=x³, y=, y=, y=

Exponential Function

y=(a>0, a≠1), a>1 monotonically increments, 0

Logarithmic function

y=, a>1 monotonically incremented, 0

Trigonometric function

y=sin x, x∈(-∞, ∞), y∈[-1, 1], T=2π

y=cos x, x∈(-∞, ∞), y∈[-1, 1], T=2π

y=tan x, x∈{x|x≠kπ±, k∈Z}, y∈(-∞, ∞), T=π

y=cot x, x∈{x|x≠kπ, k∈Z}, y∈(-∞, ∞), T=π

y=sec x=, x∈{x|x≠kπ±,k∈Z}, y∈(-∞,-1]∪[1, ∞), T=2π

y=csc x=, x∈{x|x≠kπ, k∈Z}, y∈(-∞,-1]∪[1, ∞), T=2π

Inverse trigonometric function

y=arcsin x, x∈[-1,1], y∈[-,]

y=arccos x, x∈[-1, 1], y∈[0,]

y=arctan x, x∈(-∞,∞), y∈(-,)

y=arccot x, x∈(-∞,∞), y∈(0,)

Elementary function definition

A function composed of a constant and a basic elementary function through finite four operations and a finite compound operation and can be represented by one equation.

function

Function definition

Suppose x and y are two variables (both values are taken in the real set R), and D is a given set of non-null numbers. If for each number x∈D, according to a corresponding rule f, variable y has a unique and definite value corresponding to it, then variable y is called a function of variable x, and is denoted as y=f(x)

D is called the domain of function y=f(x), x is called the independent variable, y is called the dependent variable, and the set composed of the entire function value f(x) is called the domain of the function

Properties of functions

Boundary

Let y=f(x) be defined in interval A

If there is a positive number M, for any x∈A, constant|f(x)|≤M, then y=f(x) is bounded on interval A; otherwise, it is called unbounded

If there is a real number, for any x∈A, there is always f(x)≤, then y=f(x) is said to have an upper bound on interval A

If there is a real number, for any x∈A, there is always f(x)≥, then y=f(x) is said to have a lower bound on interval A

y=f(x) has boundary on interval A ↔ both upper and lower boundary

Monotony

Let y=f(x) be defined in interval A...

Then y=f(x) is monotonically increasing (or monotonically decreasing) in interval A

Periodicity

Suppose the definition domain of f(x) is D. If there is a constant T that is not zero, such that for any x∈D, there is x±T∈D and f(x±T)=f(x), then f(x) is called a period function, and T is called a period of f(x)

Usually, the minimum positive number T satisfying the above formula is called the period that f(x)

Parity

Suppose the definition domain D of f(x) is symmetric about the origin. If for any x∈D, there is always f(-x)=f(x) (or f(-x)=-f(x)), then f(x) is called an even function (or odd function)

The graph of even functions is symmetric about the y-axis, and the graph of odd functions is symmetric about the origin

Compound functions

Suppose y=f(u), u=φ(x), if the value range of φ(x) and the definition domain of f(u) have non-empty intersection

Then the composite function y=f(u) and u=φ(x) can be compounded. The composite function y=f[φ(x)] is called the intermediate variable.

Inverse function

Suppose the definition domain of y=f(x) is D and the value domain is W

If y∈W, there is a uniquely determined x∈D, satisfying y=f(x)

Then the resulting x is a function of y, denoted as x=φ(y), called the inverse function of y=f(x), and the habit is denoted as y=

Implicit functions

There is a relational formula F(x,y)=0. If x∈D, there is a uniquely determined y satisfies F(x,y)=0 corresponding to x

The function relationship between y and x determined by this is y=y(x) is called the hidden function determined by the equation F(x,y)=0.