Function - Revision history

Rongzhou: /* Formal definition */

2026-10-16T12:07:54Z

Formal definition

← Older revision		Revision as of 12:07, 16 October 2026
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	$$		$$

	then function $f$ is the '''bijection''' of $D$ to $R_f$.		then function $f$ is said to be '''reversible''' and the '''bijection''' of $D$ to $R_f$. $f$ and $f^{-1}$ are said to be '''inverse functions''' of each other.

	<blockquote>		<blockquote>

Rongzhou at 12:05, 16 October 2026

2026-10-16T12:05:19Z

← Older revision		Revision as of 12:05, 16 October 2026
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	A '''function''' is an one~~-to-one or many~~-to-one map between two [[set\|sets]].		A '''function''' is a many(one)-to-one map between two [[set\|sets]].

	== Formal definition ==		== Formal definition ==

Rongzhou at 12:04, 16 October 2026

2026-10-16T12:04:23Z

← Older revision		Revision as of 12:04, 16 October 2026
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	A '''function''' is a many-to-one map between two [[set\|sets]].		A '''function''' is an one-to-one or many-to-one map between two [[set\|sets]].

	== Formal definition ==		== Formal definition ==

Rongzhou: /* Formal definition */

2026-10-16T12:03:44Z

Formal definition

← Older revision		Revision as of 12:03, 16 October 2026
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	== Formal definition ==		== Formal definition ==

	Let $D,R \neq \emptyset$ be [[set\|sets]].		Let $D_f,C_f,R_f \neq \emptyset$ be [[set\|sets]].

	If		If

	$$		$$
	\forall x \in D: \exists y \in R, f(x)=y		\forall x \in D_f: \exists y \in C_f, f(x)=y,
	$$		$$

	~~<ref>~~{{~~#cite:Q2199}~~}~~</ref>~~		then $f$ is a function of '''domain''' $D_f$ and '''codomain''' $C_f$, noted as $f: D_f \to C_f$;

			Furthermore,

			if

			$$
			\forall y \in R_f \subseteq C_f, \exists x \in D_f, f(x)=y,
			$$

			then $R_f$ is the '''range''' of $f$;

			Lastly,

			if

			$$
			\forall y \in R_f: \exists f^{-1}, f^{-1}(y)=x,
			$$

			then function $f$ is the '''bijection''' of $D$ to $R_f$.

	~~Then $f$~~ is a function of ~~'''~~domain~~''' $D$ and '''~~range~~''' $R$, noted as $f: D \to R$~~.		<blockquote>
			While a function can be many-to-one, a bijection is an one-to-one function of its domain to its range.
			</blockquote>

	'''Example''':		'''Example''':

	$$		$$
	f(x)=2x^2, x \in \mathbb{C}		f(x)=2x^2, x \in \mathbb{R}
	$$		$$

	is a function, where $f: \mathbb{C} \to \mathbb{C}$		is a function, where $f: \mathbb{R} \to \mathbb{R}$.

			It is however not a bijection from $\mathbb{R}$ to $\mathbb{R}$, as $\exists y \in \mathbb{R}, \nexists f^{-1}, f^{-1}(y)=x$, such as $y=-1$

			<blockquote>
			Note the range $R_f \subset \mathbb{R}$, as $\exists y \in \mathbb{R}: y \notin R_f$, such as $y=-1$
			</blockquote>

			<ref>{{#cite:Q2199}}</ref>

	== Image and preimage ==		== Image and preimage ==

Rongzhou: /* Image and preimage */

2026-10-16T11:43:49Z

Image and preimage

← Older revision		Revision as of 11:43, 16 October 2026
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	$$		$$

	Then $B_1$ is known as the '''image''' of $A_1$ in $B$ under $f$, and $A_1$ is known as the '''pre-image''' of $B_1$ under $f$.		Then $B_1$ is known as the '''image''' of $A_1$ in $B$ under $f$, and $A_1$ is known as the '''pre-image''' of $B_1$ under $f$, noted $f^{\leftarrow}(B_1)=A_1$

	<ref>{{#cite:Q2199}}</ref>		<ref>{{#cite:Q2199}}</ref>

Rongzhou: /* Formal definition */

2026-10-16T11:32:29Z

Formal definition

← Older revision		Revision as of 11:32, 16 October 2026
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	\forall x \in D: \exists y \in R, f(x)=y		\forall x \in D: \exists y \in R, f(x)=y
	$$		$$

			<ref>{{#cite:Q2199}}</ref>

	Then $f$ is a function of '''domain''' $D$ and '''range''' $R$, noted as $f: D \to R$.		Then $f$ is a function of '''domain''' $D$ and '''range''' $R$, noted as $f: D \to R$.

Rongzhou: /* Image and preimage */

2026-10-16T11:32:11Z

Image and preimage

← Older revision		Revision as of 11:32, 16 October 2026
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	is a function, where $f: \mathbb{C} \to \mathbb{C}$		is a function, where $f: \mathbb{C} \to \mathbb{C}$

			== Image and preimage ==

			Given function $f: A \to B$

			If for $A_1 \subseteq A, B_1 \subseteq B$,

			$$
			B_1=f(A_1)=\{f(a)\|\forall a \in A_1\}
			$$

			Then $B_1$ is known as the '''image''' of $A_1$ in $B$ under $f$, and $A_1$ is known as the '''pre-image''' of $B_1$ under $f$.

			<ref>{{#cite:Q2199}}</ref>

Rongzhou: Created page with "A '''function''' is a many-to-one map between two sets. == Formal definition == Let $D,R \neq \emptyset$ be sets. If $$ \forall x \in D: \exists y \in R, f(x)=y $$ Then $f$ is a function of '''domain''' $D$ and '''range''' $R$, noted as $f: D \to R$. '''Example''': $$ f(x)=2x^2, x \in \mathbb{C} $$ is a function, where $f: \mathbb{C} \to \mathbb{C}$"

2026-10-16T11:25:58Z

Created page with "A '''function''' is a many-to-one map between two sets. == Formal definition == Let $D,R \neq \emptyset$ be sets. If $$ \forall x \in D: \exists y \in R, f(x)=y $$ Then $f$ is a function of '''domain''' $D$ and '''range''' $R$, noted as $f: D \to R$. '''Example''': $$ f(x)=2x^2, x \in \mathbb{C} $$ is a function, where $f: \mathbb{C} \to \mathbb{C}$"

New page

A '''function''' is a many-to-one map between two [[set|sets]].

== Formal definition ==

Let $D,R \neq \emptyset$ be [[set|sets]].

If

$$
\forall x \in D: \exists y \in R, f(x)=y
$$

Then $f$ is a function of '''domain''' $D$ and '''range''' $R$, noted as $f: D \to R$.

'''Example''':

$$
f(x)=2x^2, x \in \mathbb{C}
$$

is a function, where $f: \mathbb{C} \to \mathbb{C}$