This is a beta course, so its structure, chapters, and examples may continue to change.

Composite Functions

Definition

Definition of Composite Functions

If $y = f(u)$ , and $u = g(x)$ , then $y = f[g(x)]$ is called the composite function formed by $f$ and $g$ , denoted as $f \circ g$ .

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Document
$f(u)$	Mathematical Symbol	f of u	Outer function, with u as the independent variable
$g(x)$	Mathematical Symbol	g of x	Inner function, with x as the independent variable
$f[g(x)]$	Mathematical Symbol	f of g of x	Composite function, composition of f and g
$f \circ g$	Mathematical Symbol	f circle g	Composite function notation, read as “f composed with g”

Properties

Composite functions have the following important properties:

Domain: The domain of a composite function is the set of x such that $g(x)$ is defined and $f(g(x))$ makes sense
Commutative Law: Function composition does not satisfy the commutative law: $f \circ g \neq g \circ f$
Associative Law: Function composition satisfies the associative law: $(f \circ g) \circ h = f \circ (g \circ h)$

Examples

Common examples of composite functions:

$f(x) = \sin(x^2)$ is the composite of $f(u) = \sin u$ and $g(x) = x^2$
$f(x) = e^{\sqrt{x}}$ is the composite of $f(u) = e^u$ and $g(x) = \sqrt{x}$
$f(x) = \ln(x^2 + 1)$ is the composite of $f(u) = \ln u$ and $g(x) = x^2 + 1$

Methods for Finding the Domain of Composite Functions

First Step: Find the domain of the inner function $g(x)$
Second Step: Find the domain of the outer function $f(u)$
Third Step: Find the range of x such that the value of $g(x)$ belongs to the domain of $f(u)$

Derivative of Composite Functions

The derivative of composite functions can be found using the chain rule:

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Exercises

Exercise 1

Find the domain of the composite function $f(x) = \ln(\sqrt{x^2 + 1})$ .

Reference Answer(4 个标签)

composite functiondomainlogarithmic functionradical function

Solution Approach: We need to consider the domains of the inner function $\sqrt{x^2 + 1}$ and the outer function $\ln u$ respectively.

Detailed Steps:

Domain of the inner function $\sqrt{x^2 + 1}$ : $x^2 + 1 \geq 0$ , which holds for all real numbers x
Domain of the outer function $\ln u$ : $u > 0$ , i.e., $\sqrt{x^2 + 1} > 0$
Since $x^2 + 1 \geq 1 > 0$ , so $\sqrt{x^2 + 1} > 0$ holds for all real numbers x

Answer: The domain is $\mathbb{R}$ (all real numbers).

$\mathbb{R}$ (blackboard bold R): This is the standard mathematical symbol for the set of real numbers (Real numbers), i.e., the set of all real numbers. Blackboard bold is a special font style used in mathematics to represent sets, to distinguish set symbols from ordinary variables.

Exercise 2

Find the domain of the composite function $f(x) = \sin(\ln x)$ .

Reference Answer(4 个标签)

composite functiondomaintrigonometric functionlogarithmic function

Solution Approach: We need to consider the domains of the inner function $\ln x$ and the outer function $\sin u$ respectively.

Detailed Steps:

Domain of the inner function $\ln x$ : $x > 0$
Domain of the outer function $\sin u$ : $u \in \mathbb{R}$ , defined for all real numbers
Therefore the domain of the composite function is the domain of the inner function

Answer: The domain is $(0, +\infty)$ .

Exercise 3

Find the domain of the composite function $f(x) = \sqrt{\frac{1}{x-1}}$ .

Reference Answer(4 个标签)

composite functiondomainradical functionrational function

Solution Approach: We need to consider the domains of the inner function $\frac{1}{x-1}$ and the outer function $\sqrt{u}$ respectively.

Detailed Steps:

Domain of the inner function $\frac{1}{x-1}$ : $x \neq 1$
Domain of the outer function $\sqrt{u}$ : $u \geq 0$ , i.e., $\frac{1}{x-1} \geq 0$
Solve the inequality $\frac{1}{x-1} \geq 0$ $\frac{1}{x - 1} \geq 0$ :
- When $x-1 > 0$ , $\frac{1}{x-1} > 0 \geq 0$ , which holds
- When $x-1 < 0$ , $\frac{1}{x-1} < 0$ , which does not hold
So $x-1 > 0$ , i.e., $x > 1$

Answer: The domain is $(1, +\infty)$ .

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
$f(u)$	Mathematical Symbol	f of u	Outer function, with u as the independent variable
$g(x)$	Mathematical Symbol	g of x	Inner function, with x as the independent variable
$f[g(x)]$	Mathematical Symbol	f of g of x	Composite function, composition of f and g
$f \circ g$	Mathematical Symbol	f composed with g	Composite function notation
$f'(x)$	Mathematical Symbol	f prime of x	First derivative of the function
$f'(g(x))$	Mathematical Symbol	f prime of g of x	Derivative of the outer function at the value of the inner function
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, the set of all real numbers
$(a, +\infty)$	Mathematical Symbol	Open interval	Left-open right-infinite interval

Chinese-English Glossary

Chinese Term	English Term	Pronunciation	Explanation
复合函数	composite function	/ˈkɒmpəzɪt ˈfʌŋkʃən/	A function formed by nesting two or more functions
外层函数	outer function	/ˈaʊtə ˈfʌŋkʃən/	The function at the outer layer in a composite function
内层函数	inner function	/ˈɪnə ˈfʌŋkʃən/	The function at the inner layer in a composite function
交换律	commutative law	/kəˈmjuːtətɪv lɔː/	The property that operations satisfy commutativity
结合律	associative law	/əˈsəʊʃɪətɪv lɔː/	The property that operations satisfy associativity
链式法则	chain rule	/tʃeɪn ruːl/	The rule for differentiating composite functions
嵌套	nesting	/ˈnestɪŋ/	Multi-layer combination of functions

PreviousSpecial Function FormsComposite, inverse, piecewise, implicit, and parametric functions—definitions, quick methods, and practice.

NextInverse Functions学习反函数的定义、性质和求法，掌握反函数与原函数的关系。