6.1: Functions

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A function from a set $A$ into a set $B$ is a relation from $A$ into $B$ such that each element of $A$ is related to exactly one element of the set $B$. The set $A$ is called the domain of the function, and the set $B$ is called the co-domain. Functions are fundamental in both mathematics and computer science for describing mathematical relationships and implementing computational logic.

In Sage, functions can be defined using direct definition.

For example, defining a function $f: \mathbb{R} \rightarrow \mathbb{R}$ to calculate the cube of a number, such as 3:

f(x) = x^3
show(f)
f(3)

\begin{matrix} (6.1.1) & x \mapsto x^{3} \end{matrix}

$/*<![CDATA[*/\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{3}/*]]>*/$

Graphical Representations

Sage provides powerful tools for visualizing functions, enabling you to explore the graphical representations of mathematical relationships.

For example, to plot the function $f(x)=x^3$ over the interval :[−2,2]:

f(x) = x^3
plot(f(x), x, -2, 2)

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Graphical Representations