Functions

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Function”,“label”:“function”,“_index”:0}] Definition 1 (Function). A function $f$ is a set of ordered pairs $(x,y)$ with $x,y\in \mathbb{R}$ such that no two ordered pairs have the same first number.

• $y$ determined by $x$ $⟹$ $y=f(x)$
• $f:X\to Y$, $X$ is domain, $Y$ is range or superset of range

Injection §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Injective”,“label”:“injective”,“_index”:1}] Definition 2 (Injective). $f$ is injective if $\forall y\in Y,\exists !x\mid f(x)=y$.

• notice we can also think of this as one to one from y to x
  • one y has one x

Surjection §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Surjection”,“label”:“surjection”,“_index”:2}] Definition 3 (Surjection). $f:X\to Y$ is surjective if $Y$ is the image of $f$, i.e. for every $y\in Y$, there is $x\in X\mid f(x)=y$.

• notice that this is “onto” from y to x
  • every $y$ has at least one $x$.

Bijection §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Bijection”,“label”:“bijection”,“_index”:3}] Definition 4 (Bijection). $f$ is bijective if it is injective and surjective.

• intuitively, this is “one-to-one” both ways

Intervals §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Neighborhod”,“label”:“neighborhod”,“_index”:4}] Definition 5 (Neighborhod). Any open interval with $a$ as its midpoint is a neighborhood of $a$.

Extremum §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Extremum”,“label”:“extremum”,“_index”:5}] Definition 6 (Extremum). $f$ has an absolute maximum on a set $S$ if there is at least one $c\in S\mid f(x)\le f(c)$ for all $x\in S$. Absolute minimum is defined similarly. An extremum is an absolute max or min.