Properties of Functions

Standard functions

There are a number of standard functions that occur frequently:

For every set $X$ , there is a unique function, called the empty function from the empty set to $X$ . The graph of an empty function is the empty set. The existence of the empty function is a convention that is needed for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements.
For every set $X$ and every singleton set {s}, there is a unique function from $X$ to {s}, which maps every element of $X$ to $s$ . This is a surjection (see below) unless $X$ is the empty set.
Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon X\to Y,} the canonical surjection of $f$ onto its image Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X)=\{f(x)\mid x\in X\}} is the function from $X$ to $f (X)$ that maps $x$ to $f (x)$ .
For every subset $A$ of a set $X$ , the inclusion map of $A$ into $X$ is the injective (see below) function that maps every element of $A$ to itself.
The identity function on a set $X$ , often denoted by $id X$ , is the inclusion of $X$ into itself.

Function composition

Given two functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon X\to Y} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\colon Y\to Z} such that the domain of $g$ is the codomain of $f$ , their composition is the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \circ f\colon X \rightarrow Z} defined by

(g\circ f)(x)=g(f(x)).

That is, the value of $g\circ f$ is obtained by first applying $f$ to $x$ to obtain $y = f (x)$ and then applying $g$ to the result $y$ to obtain $g (y) = g (f (x))$ . In the notation the function that is applied first is always written on the right.

The composition $g\circ f$ is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both $g\circ f$ and $f\circ g$ satisfy these conditions, the composition is not necessarily commutative, that is, the functions $g\circ f$ and $f\circ g$ need not be equal, but may deliver different values for the same argument. For example, let $f (x) = x 2$ and $g (x) = x + 1$ , then $g(f(x))=x^{2}+1$ and $f(g(x))=(x+1)^{2}$ agree just for $x=0.$

The function composition is associative in the sense that, if one of $(h\circ g)\circ f$ and $h\circ (g\circ f)$ is defined, then the other is also defined, and they are equal. Thus, one writes

h\circ g\circ f=(h\circ g)\circ f=h\circ (g\circ f).

The identity functions $\operatorname {id} _{X}$ and $\operatorname {id} _{Y}$ are respectively a right identity and a left identity for functions from $X$ to $Y$ . That is, if $f$ is a function with domain $X$ , and codomain $Y$ , one has $f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.$

A composite function g(f(x)) can be visualized as the combination of two "machines".
A simple example of a function composition
Another composition. In this example, $(g \circ f)(c) = #$ .

Image and preimage

Let $f\colon X\to Y.$ The image under $f$ of an element $x$ of the domain $X$ is $f (x)$ . If $A$ is any subset of $X$ , then the image of $A$ under $f$ , denoted $f (A)$ , is the subset of the codomain $Y$ consisting of all images of elements of $A$ , that is,

f(A)=\{f(x)\mid x\in A\}.

The image of $f$ is the image of the whole domain, that is, $f (X)$ . It is also called the range of $f$ , although the term range may also refer to the codomain.

On the other hand, the inverse image or preimage under $f$ of an element $y$ of the codomain $Y$ is the set of all elements of the domain $X$ whose images under $f$ equal $y$ . In symbols, the preimage of $y$ is denoted by $f^{-1}(y)$ and is given by the equation

f^{-1}(y)=\{x\in X\mid f(x)=y\}.

Likewise, the preimage of a subset $B$ of the codomain $Y$ is the set of the preimages of the elements of $B$ , that is, it is the subset of the domain $X$ consisting of all elements of $X$ whose images belong to $B$ . It is denoted by $f^{-1}(B)$ and is given by the equation

f^{-1}(B)=\{x\in X\mid f(x)\in B\}.

For example, the preimage of $\{4,9\}$ under the square function is the set $\{-3,-2,2,3\}$ .

By definition of a function, the image of an element $x$ of the domain is always a single element of the codomain. However, the preimage $f^{-1}(y)$ of an element $y$ of the codomain may be empty or contain any number of elements. For example, if $f$ is the function from the integers to themselves that maps every integer to 0, then $f^{-1}(0)=\mathbb {Z}$ .

If $f\colon X\to Y$ is a function, $A$ and $B$ are subsets of $X$ , and $C$ and $D$ are subsets of $Y$ , then one has the following properties:

$A\subseteq B\Longrightarrow f(A)\subseteq f(B)$
$C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)$
$A\subseteq f^{-1}(f(A))$
$C\supseteq f(f^{-1}(C))$
$f(f^{-1}(f(A)))=f(A)$
$f^{-1}(f(f^{-1}(C)))=f^{-1}(C)$

The preimage by $f$ of an element $y$ of the codomain is sometimes called, in some contexts, the fiber of $y$ under $f$ .

If a function $f$ has an inverse (see below), this inverse is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}.} In this case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(C)} may denote either the image by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} or the preimage by $f$ of $C$ . This is not a problem, as these sets are equal. The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(C)} may be ambiguous in the case of sets that contain some subsets as elements, such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x, \{x\}\}.} In this case, some care may be needed, for example, by using square brackets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f[A], f^{-1}[C]} for images and preimages of subsets and ordinary parentheses for images and preimages of elements.

Injective, surjective and bijective functions

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon X\to Y} be a function.

The function $f$ is injective (or one-to-one, or is an injection) if $f (a) \neq f (b)$ for any two different elements $a$ and $b$ of $X$ . Equivalently, $f$ is injective if and only if, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in Y,} the preimage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} contains at most one element. An empty function is always injective. If $X$ is not the empty set, then $f$ is injective if and only if there exists a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\colon Y\to X} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\circ f=\operatorname{id}_X,} that is, if $f$ has a left inverse. Proof: If $f$ is injective, for defining $g$ , one chooses an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} in $X$ (which exists as $X$ is supposed to be nonempty), and one defines $g$ by $g(y)=x$ if $y=f(x)$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(y)=x_0} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\not\in f(X).} Conversely, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\circ f=\operatorname{id}_X,} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x),} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=g(y),} and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)=\{x\}.}

The function $f$ is surjective (or onto, or is a surjection) if its range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X)} equals its codomain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , that is, if, for each element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} of the codomain, there exists some element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} of the domain such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = y} (in other words, the preimage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} of every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in Y} is nonempty). If, as usual in modern mathematics, the axiom of choice is assumed, then $f$ is surjective if and only if there exists a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\colon Y\to X} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ g=\operatorname{id}_Y,} that is, if $f$ has a right inverse. The axiom of choice is needed, because, if $f$ is surjective, one defines $g$ by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(y)=x,} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is an arbitrarily chosen element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y).}

The function $f$ is bijective (or is a bijection or a one-to-one correspondence) if it is both injective and surjective. That is, $f$ is bijective if, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in Y,} the preimage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} contains exactly one element. The function $f$ is bijective if and only if it admits an inverse function, that is, a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\colon Y\to X} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\circ f=\operatorname{id}_X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ g=\operatorname{id}_Y.} (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).

Every function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon X\to Y} may be factorized as the composition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\circ s} of a surjection followed by an injection, where $s$ is the canonical surjection of $X$ onto $f (X)$ and $i$ is the canonical injection of $f (X)$ into $Y$ . This is the canonical factorization of $f$ .

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " $f$ maps $X$ onto $Y$ " differs from " $f$ maps $X$ into $B$ ", in that the former implies that $f$ is surjective, while the latter makes no assertion about the nature of $f$ . In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.

Restriction and extension

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon X \to Y} is a function and S is a subset of X, then the restriction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} to S, denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_S} , is the function from S to Y defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_S(x) = f(x)}

for all x in S. Restrictions can be used to define partial inverse functions: if there is a subset S of the domain of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_S} is injective, then the canonical surjection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_S} onto its image Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_S(S) = f(S)} is a bijection, and thus has an inverse function from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(S)} to S. One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval [0, π]. The image of this restriction is the interval [−1, 1], and thus the restriction has an inverse function from [−1, 1] to [0, π], which is called arccosine and is denoted $arccos$ .

Function restriction may also be used for "gluing" functions together. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle X=\bigcup_{i\in I}U_i} be the decomposition of $X$ as a union of subsets, and suppose that a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i\colon U_i \to Y} is defined on each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} such that for each pair Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i, j} of indices, the restrictions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_j} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i \cap U_j} are equal. Then this defines a unique function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon X \to Y} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_{U_i} = f_i} for all $i$ . This is the way that functions on manifolds are defined.

An extension of a function $f$ is a function $g$ such that $f$ is a restriction of $g$ . A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying homographies of the real line. A homography is a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)=\frac{ax+b}{cx+d}} such that $ad - bc \neq 0$ . Its domain is the set of all real numbers different from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -d/c,} and its image is the set of all real numbers different from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a/c.} If one extends the real line to the projectively extended real line by including $\infty$ , one may extend $h$ to a bijection from the extended real line to itself by setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(\infty)=a/c} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(-d/c)=\infty} .

Resources

Function and Function Notation,Book Chapter
Function and Function Notation Guided Notes

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Properties of Functions

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Standard functions

Function composition

Image and preimage

Injective, surjective and bijective functions

Restriction and extension

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