Function: Difference between revisions
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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 $f: \mathbb{C} \to \mathbb{C}$" |
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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> | |||
Revision as of 11:32, 16 October 2026
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 $f: \mathbb{C} \to \mathbb{C}$
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 $f$.