Functions, Domain, and Range: Precalculus Study Notes

Functions and Their Properties

Definition of a Function

A function is a relationship between two sets, typically called the domain and the range, where each input from the domain is assigned exactly one output in the range.

• Domain: The set of all possible input values (usually represented by x).

• Range: The set of all possible output values (usually represented by y).

• Independent Variable: The variable representing the input (commonly x).

• Dependent Variable: The variable representing the output (commonly y).

A function is often written as $y = f(x)$, which is read as "y equals f of x."

Identifying Functions from Tables

To determine if a table represents a function, check that each input value corresponds to exactly one output value.

x	y
1	2
2	3
3	9
4	18
5	18

In this table, each x-value has only one corresponding y-value, so it represents a function.

x	y
1	5
1	3
2	3
4	8
5	16

Here, the input x = 1 corresponds to two different outputs (5 and 3), so this is not a function.

Vertical Line Test

The vertical line test is a graphical method to determine if a curve is a function. If any vertical line crosses the graph more than once, the graph does not represent a function.

• If every vertical line intersects the graph at most once, the graph is a function.

• If a vertical line intersects the graph more than once, it is not a function.

Evaluating Functions

Function Notation and Evaluation

Given a function $f(x)$, you can find the output for any input by substituting the value of x into the function.

• Example 1: If $f(x) = x - 2$, then $f(-5) = -5 - 2 = -7$ and $f(1) = 1 - 2 = -1$.

• Example 2: If $f(x) = (x - 2)^2$, then $f(-2) = (-2 - 2)^2 = 16$ and $f(0) = (0 - 2)^2 = 4$.

• Example 3: If $f(y) = y - 1$, then $f(2) = 2 - 1 = 1$ and $f(x+3) = (x+3) - 1 = x + 2$.

Domain and Range

Definitions

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) that the function can produce.

Interval Notation and Inequalities

Domains and ranges are often expressed using interval notation or inequality notation.

Inequality Symbol	Interval Notation	Plotting on # Line	Graphing Inequality
< or >	( )	Open circle	Dotted line
≤ or ≥	[ ]	Closed circle	Solid line

Infinity ($\infty$) is always paired with a parenthesis, never a bracket, because infinity is not a specific number.

Common Inequality and Interval Notations

Inequality	Interval Notation
$a < x < b$	$(a, b)$
$a \leq x \leq b$	$[a, b]$
$a < x \leq b$	$(a, b]$
$a \leq x < b$	$[a, b)$
$x < a$	$(-\infty, a)$
$x \leq a$	$(-\infty, a]$
$x > b$	$(b, \infty)$
$x \geq b$	$[b, \infty)$
$x < a$ or $x > b$	$(-\infty, a) \cup (b, \infty)$
$x \leq a$ or $x \geq b$	$(-\infty, a] \cup [b, \infty)$

Types of Functions and Their Domains

Common Function Types

• Polynomial Functions: $f(x) = ax^n + bx^{n-1} + \ldots + c$

• Power Functions: $f(x) = x^n$ (n even or odd)

• Root Functions: $f(x) = \sqrt[n]{x}$

• Reciprocal Functions: $f(x) = \frac{1}{x^n}$

• Absolute Value Function: $f(x) = |x|$

• Greatest Integer Function: $f(x) = \lfloor x \rfloor$

Domains of Common Functions

Function	Domain: Interval Notation	Domain: Inequality
Polynomial	$(-\infty, \infty)$	$x \in \mathbb{R}$
Power; n is even	$(-\infty, \infty)$	$x \in \mathbb{R}$
Power; n is odd	$(-\infty, \infty)$	$x \in \mathbb{R}$
Root; n is even	$[0, \infty)$	$x \geq 0$
Root; n is odd	$(-\infty, \infty)$	$x \in \mathbb{R}$
Reciprocal; n is even	$(-\infty, 0) \cup (0, \infty)$	$x < 0$ or $x > 0$
Reciprocal; n is odd	$(-\infty, 0) \cup (0, \infty)$	$x < 0$ or $x > 0$
Reciprocal of root; n is even	$(0, \infty)$	$x > 0$
Reciprocal of root; n is odd	$(-\infty, 0) \cup (0, \infty)$	$x < 0$ or $x > 0$
Absolute Value	$(-\infty, \infty)$	$x \in \mathbb{R}$

Graphical Representations of Functions

Common functions have characteristic graphs. For example:

• Linear functions: Straight lines ($f(x) = mx + b$)

• Quadratic functions: Parabolas ($f(x) = ax^2 + bx + c$)

• Cubic functions: S-shaped curves ($f(x) = ax^3 + bx^2 + cx + d$)

• Square root functions: Half-parabola shapes ($f(x) = \sqrt{x}$)

• Reciprocal functions: Hyperbolas ($f(x) = \frac{1}{x}$)

• Absolute value functions: V-shaped graphs ($f(x) = |x|$)

Understanding the domain and range of each function type is essential for graphing and solving equations.

Example: Finding Domain and Range from a Graph

• To find the domain, look for the set of all x-values covered by the graph.

• To find the range, look for the set of all y-values the graph attains.

• Use interval notation to express both domain and range.

Additional info: The notes also include blank tables for students to practice writing inequalities and interval notation, as well as a summary chart of function types and their domains. The graphical representations and vertical line test are standard tools in Precalculus for understanding functions.