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Inverse Functions

Definition

Definition of the Inverse Function

If $y = f(x)$ is strictly monotonic, then it is one-to-one, so an inverse function exists, written $x = f^{-1}(y)$ or $y = f^{-1}(x)$ . In general, a function has an inverse if and only if it is one-to-one when viewed as a map onto its range.

Properties

An inverse function has the following important properties:

Symmetry: The graphs of a function and its inverse are symmetric about the line $y = x$
Domain: The domain of the inverse is the range of the original function
Range: The range of the inverse is the domain of the original function
Inverse of inverse: $(f^{-1})^{-1} = f$ (the inverse of the inverse is the original function)
Monotonicity: The inverse has the same monotonicity as the original function

How to Find an Inverse Function

Step 1: Solve $y = f(x)$ for $x$ in terms of $y$
Step 2: Swap $x$ and $y$ to get the usual notation of the inverse
Step 3: Determine the domain of the inverse (that is, the range of the original function)

Inverses of Common Functions

Original function	Inverse	Domain
$y = x^2$ ( $x \geq 0$ )	$y = \sqrt{x}$	$[0, +\infty)$
$y = e^x$	$y = \ln x$	$(0, +\infty)$
$y = \sin x$ ( $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ )	$y = \arcsin x$	$[-1, 1]$
$y = \cos x$ ( $x \in [0, \pi]$ )	$y = \arccos x$	$[-1, 1]$
$y = \tan x$ ( $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ )	$y = \arctan x$	$\mathbb{R}$

Derivative of an Inverse Function

If $y = f^{-1}(x)$ , then:

$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$

Exercises

Exercise 1

Find the inverse of $y = 2x + 1$ .

Answer and Explanation(2 个标签)

inverse functionlinear function

Approach: Follow the steps for finding an inverse function.

Detailed steps:

Solve $y = 2x + 1$ for x: $y - 1 = 2x$ $x = \frac{y - 1}{2}$
Swap x and y: $y = \frac{x - 1}{2}$
Determine the domain: The range of $y = 2x + 1$ is $\mathbb{R}$ , so the domain of the inverse is also $\mathbb{R}$ .

Answer: The inverse is $y = \frac{x - 1}{2}$ , with domain $\mathbb{R}$ .

Exercise 2

Find the inverse of $y = x^2 + 2$ ( $x \geq 0$ ).

Answer and Explanation(3 个标签)

inverse functionquadratic functiondomain

Approach: Follow the steps for finding an inverse function, paying attention to the restriction on the domain.

Detailed steps:

Solve $y = x^2 + 2$ for x: $y - 2 = x^2$ $x = \sqrt{y - 2}$ (since $x \geq 0$ )
Swap x and y: $y = \sqrt{x - 2}$
Determine the domain: The range of $y = x^2 + 2$ ( $x \geq 0$ ) is $[2, +\infty)$ , so the domain of the inverse is $[2, +\infty)$ .

Answer: The inverse is $y = \sqrt{x - 2}$ , with domain $[2, +\infty)$ .

Exercise 3

Find the inverse of $y = \frac{1}{x}$ ( $x \neq 0$ ).

Answer and Explanation(3 个标签)

inverse functionrational functionself-inverse

Approach: Follow the steps for finding an inverse function.

Detailed steps:

Solve $y = \frac{1}{x}$ for x: $xy = 1$ $x = \frac{1}{y}$
Swap x and y: $y = \frac{1}{x}$
Determine the domain: The range of $y = \frac{1}{x}$ ( $x \neq 0$ ) is $\mathbb{R} \setminus \{0\}$ , so the domain of the inverse is also $\mathbb{R} \setminus \{0\}$ .

Answer: The inverse is $y = \frac{1}{x}$ , with domain $\mathbb{R} \setminus \{0\}$ .

Note: This function is its own inverse; such a function is called a self-inverse function.

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$f(x)$	Math symbol	f of x	A function of variable $x$
$f^{-1}(x)$	Math symbol	f inverse of x	The inverse of the function $f(x)$
$y = f(x)$	Math symbol	y equals f of x	General form of a function
$\mathbb{R}$	Math symbol	Real numbers	The set of all real numbers
$\mathbb{R} \setminus \{0\}$	Math symbol	R minus zero	The set of real numbers excluding 0
$[a, +\infty)$	Math symbol	half-open interval	Left-closed, right-unbounded interval
$(a, +\infty)$	Math symbol	open interval	Left-open, right-unbounded interval
$\sqrt{x}$	Math symbol	square root of x	The square root of $x$
$\ln x$	Math symbol	natural logarithm of x	The natural logarithm function
$\arcsin x$	Math symbol	arc sine x	The inverse sine function
$\arccos x$	Math symbol	arc cosine x	The inverse cosine function
$\arctan x$	Math symbol	arc tangent x	The inverse tangent function

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
inverse function	反函数	/ɪnˈvɜːs ˈfʌŋkʃən/	The inverse relationship of a function
self-inverse function	自反函数	/self ɪnˈvɜːs ˈfʌŋkʃən/	A function whose inverse equals itself
monotonicity	单调性	/ˌmɒnəʊtəˈnɪsɪti/	Whether function values increase or decrease with the input
symmetry	对称性	/ˈsɪmɪtri/	A symmetry property of a function’s graph
inverse operation	逆运算	/ɪnˈvɜːs ˌɒpəˈreɪʃən/	The operation that undoes a function
chain rule	链式法则	/tʃeɪn ruːl/	The rule for differentiating composite functions

Composite Functions

Learn the definition, properties, and methods of composite functions, master the method of determining the domain of composite functions.

Piecewise Functions

Learn the definition, construction, and properties of piecewise functions, master the analysis methods of piecewise functions.