Definition and Classification of Elementary Functions

Definition of Elementary Functions

Elementary\ functions\ are\ functions\ obtained\ by\ basic\ elementary\ functions\ through\ finite\ arithmetic\ operations\ and\ composition\ operations.

Classification

Algebraic functions: Functions that only contain algebraic operations (addition, subtraction, multiplication, division, root extraction)
Transcendental functions: Functions that contain transcendental operations (exponential, logarithmic, trigonometric functions)

Examples of Algebraic Functions

Polynomial function $f(x) = x^3 - 2x + 1$
Rational function $f(x) = \frac{x^2 + 1}{x - 1}$
Irrational function $f(x) = \sqrt{x^2 + 1}$

Examples of Transcendental Functions

Exponential function $f(x) = e^x$
Logarithmic function $f(x) = \ln x$
Trigonometric function $f(x) = \sin x$
Composite function $f(x) = e^{\sin x}$

Practice Questions

Practice 1

Determine which of the following functions are elementary functions: $y = x^2$ , $y = \sin x$ , $y = \ln x$ , $y = \Gamma(x)$ .

Answer and Explanation (2 个标签)

elementary function classification

$y = x^2$ , $y = \sin x$ , $y = \ln x$ are elementary functions, $y = \Gamma(x)$ is not (Gamma function is a special function).

Practice 2

Which of the following functions are algebraic functions: $y = \sqrt{x}$ , $y = e^x$ , $y = \frac{x^2+1}{x-1}$ , $y = \sin x$ .

Answer and Explanation (1 个标签)

algebraic function

$y = \sqrt{x}$ , $y = \frac{x^2+1}{x-1}$ are algebraic functions, $y = e^x$ , $y = \sin x$ are not.

Practice 3

Which of the following functions are transcendental functions: $y = x^3$ , $y = \ln x$ , $y = \sin x$ , $y = \sqrt{x}$ .

Answer and Explanation (1 个标签)

transcendental function

$y = \ln x$ , $y = \sin x$ are transcendental functions, $y = x^3$ , $y = \sqrt{x}$ are not.

Practice 4

Determine which category of elementary functions $y = e^{\sin x}$ belongs to, and explain why.

Answer and Explanation (2 个标签)

transcendental function composite function

It belongs to transcendental functions because it contains the composition of exponential and trigonometric functions.

Practice 5

Determine which category of elementary functions $y = \sqrt{x^2 + 1}$ belongs to, and explain why.

Answer and Explanation (1 个标签)

algebraic function

It belongs to algebraic functions because it only involves algebraic operations and root extraction.

Practice 6

【2020·Mathematics One】Which of the following functions are transcendental functions?
(A) $y = x^2$
(B) $y = \sin x$
(C) $y = \ln x$
(D) $y = \sqrt{x}$

Answer and Explanation (1 个标签)

transcendental function

(B), (C) are transcendental functions, (A), (D) are not.

Practice 7

【2018·Mathematics Two】Determine whether $y = \frac{x^2 + 1}{x - 1}$ is an algebraic function, and explain why.

Answer and Explanation (1 个标签)

algebraic function

Yes, it is an algebraic function because it only involves algebraic operations such as addition, subtraction, multiplication, and division.

Practice 8

【2016·Mathematics One】Determine which category of elementary functions $y = e^{\sin x}$ belongs to, and explain why.

Answer and Explanation (1 个标签)

transcendental function

It belongs to transcendental functions because it contains the composition of exponential and trigonometric functions.

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$f(x)$	Mathematical Symbol	f of x	Function notation, representing a function with $x$ as the independent variable
$\Gamma(x)$	Mathematical Symbol	Gamma of x	Gamma function, special function, not an elementary function
$e^x$	Mathematical Symbol	e to the power x	Natural exponential function
$\ln x$	Mathematical Symbol	natural log of x	Natural logarithmic function
$\sin x$	Mathematical Symbol	sine of x	Sine function
$e^{\sin x}$	Mathematical Symbol	e to the power sine x	Composite function of exponential and trigonometric functions
$\sqrt{x}$	Mathematical Symbol	square root of x	Square root function

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
初等函数	elementary function	/ˌelɪˈmentəri ˈfʌŋkʃən/	Functions obtained by basic elementary functions through finite arithmetic and composition operations
基本初等函数	basic elementary function	/ˈbeɪsɪk ˌelɪˈmentəri ˈfʌŋkʃən/	Power functions, exponential functions, logarithmic functions, trigonometric functions, inverse trigonometric functions, etc.
代数函数	algebraic function	/ældʒɪˈbreɪɪk ˈfʌŋkʃən/	Functions that only contain algebraic operations (addition, subtraction, multiplication, division, root extraction)
超越函数	transcendental function	/trænsenˈdentəl ˈfʌŋkʃən/	Functions that contain transcendental operations (exponential, logarithmic, trigonometric functions)
四则运算	four arithmetic operations	/fɔː əˈrɪθmətɪk ˌɒpəˈreɪʃənz/	Four basic operations: addition, subtraction, multiplication, and division
复合运算	composition operation	/ˌkɒmpəˈzɪʃən ˌɒpəˈreɪʃən/	Composition operation of functions
多项式函数	polynomial function	/ˌpɒlɪˈnəʊmiəl ˈfʌŋkʃən/	Functions of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$
有理函数	rational function	/ˈræʃənəl ˈfʌŋkʃən/	Quotient of two polynomial functions
无理函数	irrational function	/ɪˈræʃənəl ˈfʌŋkʃən/	Functions containing radicals
伽马函数	gamma function	/ˈɡæmə ˈfʌŋkʃən/	Special function, not an elementary function

NextPower FunctionsDetailed explanation of the definition, properties, graphs, and typical exercises of power functions.

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