Precalculus Study Notes: Functions, Domains, and Graphs (Chapter 2, Sections 2.4–2.9)

Functions and Relations

Definition of Relations and Functions

A relation is any correspondence between the values of two variables, typically denoted as x and y, and can be described as a set of ordered pairs (x, y). The domain of a relation is the set of all possible values of x, while the range is the set of all possible values of y that correspond to the domain values.

• A function is a special type of relation in which each value of the domain corresponds to one and only one value in the range.

• If a relation assigns more than one value in the range to a single domain value, it is not a function.

Example: The relation f = {(1,4), (2,3), (3,4), (4,1)} is a function because each x-value has only one corresponding y-value.

Function Notation: For a function f, the relationship between x and y is written as f(x) = y, read as "f of x equals y."

• Independent variable: x

• Dependent variable: y

Evaluating Functions and Function Notation

Examples of Function Evaluation

• Given f(x) = |x| + 1, evaluate for x = 0 and x = -2:

  • f(0) = |0| + 1 = 1

  • f(-2) = |-2| + 1 = 2 + 1 = 3

• Given g(x) = x^2 - 2x - 3, evaluate:

  • g(0) = 0^2 - 2*0 - 3 = -3

  • g(a) = a^2 - 2a - 3

  • g(t+1) = (t+1)^2 - 2(t+1) - 3 = t^2 + 2t + 1 - 2t - 2 - 3 = t^2 - 4

Domain of a Function: The domain of f(x) is the set of all real numbers x for which f(x) is also a real number. Restrictions may occur due to division by zero or taking square roots of negative numbers.

Finding the Domain of Functions

Explicit and Implicit Domain Restrictions

• For rational functions, exclude values that cause division by zero.

• For functions involving square roots, exclude values that result in taking the square root of a negative number.

Examples:

• f(x) = \frac{2x}{x^2 - 4}

  • Set denominator ≠ 0: x^2 - 4 ≠ 0 ⇒ x ≠ ±2

  • Domain: $D = (-\infty, -2) \cup (-2, 2) \cup (2, \infty)$

• g(x) = \sqrt{2x+1}

  • Set radicand ≥ 0: 2x + 1 ≥ 0 ⇒ x ≥ -0.5

  • Domain: $D = [-0.5, \infty)$

Graphical Representation of Functions

Vertical Line Test (VLT)

The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point. This ensures that each x-value corresponds to only one y-value.

• Graphs that fail the VLT are not functions.

• Graphs that pass the VLT are functions.

Domain and Range from Graphs

Given a graph of y = f(x):

• Domain (D): The set of all x-values for which the graph exists.

• Range (R): The set of all y-values that the graph attains.

Example: For a graph with x-values from -1 to 2 and y-values from 2 to 3:

• $D = [-1, 2]$

• $R = [2, 3]$

Increasing/Decreasing Functions and Relative Max/Min Values

Definitions

• A function is increasing on an interval if its graph rises from left to right.

• A function is decreasing on an interval if its graph falls from left to right.

• A relative maximum occurs at a point where the function value is higher than at nearby points.

• A relative minimum occurs at a point where the function value is lower than at nearby points.

Example: For a function f(x) with a graph:

• f is increasing on $(-2,2)$

• f is decreasing on $(-\infty,-2)$ and $(2,\infty)$

• Relative max: $f(2) = 3$

• Relative min: $f(-2) = -3$

Even and Odd Functions

Definitions and Properties

• Even function: $f(-x) = f(x)$ for all x in the domain. The graph is symmetric about the y-axis.

• Odd function: $f(-x) = -f(x)$ for all x in the domain. The graph is symmetric about the origin.

• Functions that are neither even nor odd do not have these symmetries.

Examples:

• $f(x) = \frac{1}{1 + x^2}$ is even.

• $g(x) = \frac{x}{1 + x^2}$ is odd.

• $h(x) = x^2 - 2x$ is neither even nor odd.

Average Rate of Change of a Function

Definition and Formula

The average rate of change (AROC) of a function f on the interval [a, b] is given by:

• $\text{AROC} = \frac{f(b) - f(a)}{b - a}$

This is equivalent to the slope of the line passing through the points (a, f(a)) and (b, f(b)).

Example: For $T(t) = t^2 - 2t - 3$ on $[1, 4]$:

• $T(4) = 4^2 - 2*4 - 3 = 16 - 8 - 3 = 5$

• $T(1) = 1^2 - 2*1 - 3 = 1 - 2 - 3 = -4$

• $\text{AROC} = \frac{5 - (-4)}{4 - 1} = \frac{9}{3} = 3$

The average rate of change of temperature is 3 °C/hr.

A Library of Basic Functions

Common Functions and Their Graphs

Function Name	Formula	Graph Description
Constant Function	$f(x) = c$	Horizontal line
Identity Function	$f(x) = x$	Line through origin
Squaring Function	$f(x) = x^2$	Parabola
Square Root Function	$f(x) = \sqrt{x}$	Half of a sideways parabola
Cubing Function	$f(x) = x^3$	Steep curve with origin wiggle
Cube Root Function	$f(x) = \sqrt[3]{x}$	Sideways cubing curve
Absolute Value Function	$f(x) = |x|$	V-shaped graph
Reciprocal Function	$f(x) = \frac{1}{x}$	Two-piece curve

Summary Table: Key Properties of Functions

Property	Definition	Test/Example
Function	Each x-value maps to one y-value	Vertical Line Test
Domain	Set of all possible x-values	Exclude division by zero, negative radicands
Range	Set of all possible y-values	Project graph onto y-axis
Even Function	$f(-x) = f(x)$	y-axis symmetry
Odd Function	$f(-x) = -f(x)$	Origin symmetry
AROC	$\frac{f(b) - f(a)}{b - a}$	Slope between two points

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• All formulas are provided in LaTeX format for mathematical accuracy.

• Graphical examples are described in text due to image limitations.