Function Notation and Domain/Range

Function Notation and Domain/Range

Basic Facts about Functions

• Functions are relations, but not all relations are functions.

Notation

Set Builder Notation

• Example: The set of all x such that x is greater than -2.

Interval Notation

• Example: $(-2, \infty)$ represents all numbers greater than -2.

• Left parenthesis indicates the value is not included.

• Parenthesis is always used next to the infinity symbol.

Intervals

Open Intervals

• Round parentheses on both sides.

• Equivalent to simple inequalities.

• Example: $x > a$ (from a to infinity), $x < b$, $a < x < b$

• Value of A is not included; it approaches A but never reaches it.

Important Note

• In interval notation, the least quantity is on the left, and the greatest quantity is on the right.

Intervals with Brackets

• Bracket includes the value (greater than or equal to, or less than or equal to).

• Closed Intervals: Brackets on both sides, including both values.

Disjoint Intervals

• Connecting two intervals.

• Example: $x < a$ (from negative infinity to a) union $x > b$ (from b to infinity).

• Again, least quantity on the left, greatest on the right.

• $x$ can be any real number: $(-\infty, \infty)$.

Example

Write each set of real numbers in interval notation.

A

• The set of all $x$ such that $x \le 5$ and $x > -3$.

  $(-3, 5]$

B

• The set of all $x$ such that $x \le 3$ and the set of $x$ where $x > 3$.

• Unbounded interval.

  $(-\infty, 3] \cup (3, \infty)$.

• $x > 3$ indicates $3 < x$.

Relations, Domain, and Range

Relation

• A set of ordered pairs.

• Ordered pairs denoted by $(x, y)$.

• Domain: Set of all x values.

• Range: Set of all y values.

Example

• $F = {(1, 2), (-2, 5), (3, -1)}$
  Domain: ${1, -2, 3}$
  Range: ${2, 5, -1}$

• Graph of F will have three points.

Graphing a Relation

• The graph of a line or curve in the xy-plane represents the relation.

• Example: F represents a relation consisting of all ordered pairs having the form $(x, 2x)$, where x is a real number.

Venn Diagrams

• Domain on the left side, range on the right side.

• To find the domain, cast light towards the x-axis and see the shadow.

• To find the range, cast light towards the y-axis and see the shadow.

• Example: Domain goes from -3 to 3, and range goes from -6 to 6.

Example from Graph

• Find the domain and range of each relation from the graph.

• If casting light towards the x-axis covers the entire axis, the domain is from $(-\infty, \infty)$.

• If casting light towards the y-axis covers the entire axis, the range is from $(-\infty, \infty)$.
  Note the arrows on both sides mean it keeps going.

Another Example

• Domain is the entire number line $(-\infty, \infty)$.

• Range goes to positive infinity but stops at approximately -3.

Functions

Definition

• A function is a relation in which each element in the domain corresponds to exactly one element in the range.

Variables

• If x represents any element in the domain, then x is called the independent variable.

• If y represents any element in the range, then y is called the dependent variable (because y depends on x).

Deciding Whether Relations Are Functions

Example 1

• Relation: \left{(1, 2), (3, 4), (5, 6), (7, 8), (9, 10)\right}

• Domain: \left{1, 3, 5, 7, 9\right}

• Range: \left{2, 4, 6, 8, 10\right}

• Each element of the domain corresponds to exactly one element of the range, so this is a function.

Example 2

• Relation: \left{(1, 1), (1, 2), (1, 3), (2, 4)\right}

• Domain: \left{1, 2\right}

• Range: \left{1, 2, 3, 4\right}

• One element in the domain (1) has been assigned three different elements from the range, so this is not a function.

Example 3

• Table notation.

• Domain comes from the x's, range comes from the y's.

• If the range has only one value, it's a function.

• Every element in the domain corresponds to the same element in the range, which makes it a function.

• It's the other way around that doesn't work.

Example 4

• Set notation: Set of all x and y such that $y=x-2$.

• This is a function.

• y is always found by subtracting 2 from x.

• Each x corresponds to exactly one value of y.

• The domain and range are the set of real numbers between negative infinity and positive infinity.