College Algebra Study Guide: Functions and Their Properties

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Functions, Domain, and Range

Definition and Representation of Functions

A function is a mathematical rule that assigns each input from a set called the domain to exactly one output in a set called the range. Functions can be represented by equations, graphs, tables, or verbal descriptions.

Input (domain): The set of all possible values for the independent variable, usually denoted as x.
Output (range): The set of all possible values for the dependent variable, usually denoted as f(x).

Function input-output diagram Definition of a function Mapping diagram of domain and range

Types of Intervals

Domains and ranges are often described using intervals. Intervals can be open, closed, or half-open, and are used to specify the set of real numbers included.

Notation	Set Description	Type	Picture
(a, b)	{x \| a < x < b}	Open	Open interval
[a, b]	{x \| a ≤ x ≤ b}	Closed	Closed interval
(a, b]	{x \| a < x ≤ b}	Half-open	Half-open interval
[a, b)	{x \| a ≤ x < b}	Half-open	Half-open interval
(a, ∞)	{x \| x > a}	Open	Open interval
[a, ∞)	{x \| x ≥ a}	Closed	Closed interval
(-∞, b)	{x \| x < b}	Open	Open interval
(-∞, b]	{x \| x ≤ b}	Closed	Closed interval
ℝ	Set of all real numbers	Both open and closed	Entire real line

Types of intervals table

Domain and Range Examples

Different functions have different domains and ranges, depending on their formulas.

Function	Domain (x)	Range (y)
y = x^2	(-∞, ∞)	[0, ∞)
y = 1/x	(-∞, 0) ∪ (0, ∞)	(-∞, 0) ∪ (0, ∞)
y = √x	[0, ∞)	[0, ∞)
y = √(4 - x)	(-∞, 4]	[0, ∞)
y = √(1 - x^2)	[-1, 1]	[0, 1]

Domain and range table

Finding Domain and Range

To determine the domain and range of a function, consider restrictions such as division by zero, even roots of negative numbers, and logarithms of non-positive numbers.

Example: For each function, determine its domain and range.

Finding domain and range example

Zeros and Intercepts of Functions

The zeros of a function are the values of x for which f(x) = 0. The y-intercept is the value of the function at x = 0.

Example: For f(x) = -4x + 2, find all zeros, the y-intercept, and sketch its graph.

Finding zeros and y-intercepts example

Applications: Free-Falling Object

Functions can model real-world phenomena, such as the height of a free-falling object over time.

Example: s(t) = -16t^2 + 100 models the height of a ball dropped from 100 ft. Create a table of values, determine the domain, and sketch the graph.

Free-falling object example

Graphs of Functions

Vertical Line Test

The vertical line test is used to determine if a graph represents a function. If any vertical line intersects the graph more than once, it is not a function.

Circle graph fails vertical line test Upper semicircle graph Lower semicircle graph Vertical line test rule

Combining and Composing Functions

Combining Functions with Mathematical Operations

Functions can be added, subtracted, multiplied, or divided. The domain of the resulting function is typically the intersection of the domains of the original functions.

Graph of combined functions

Function	Formula	Domain
f + g	(f + g)(x) = √x + √(1 - x)	[0, 1]
f - g	(f - g)(x) = √x - √(1 - x)	[0, 1]
g - f	(g - f)(x) = √(1 - x) - √x	[0, 1]
f · g	(f · g)(x) = √x · √(1 - x) = √(x(1 - x))	[0, 1]
f / g	f(x) / g(x) = √x / √(1 - x)	(0, 1) (x = 1 excluded)
g / f	g(x) / f(x) = √(1 - x) / √x	(0, 1) (x = 0 excluded)

Table of combined functions

Composing Functions

The composition of two functions f and g, written as (f ∘ g)(x), means applying g first, then f to the result: .

The domain of the composition is the set of x in the domain of g for which g(x) is in the domain of f.

Composing functions example Definition of composition

Increasing and Decreasing Functions

Definitions

A function is increasing on an interval if its output grows as the input increases. It is decreasing if its output decreases as the input increases.

If whenever , then f is increasing.
If whenever , then f is decreasing.

Increasing and decreasing functions definition

Even and Odd Functions: Symmetry

Definitions

A function is even if for all x in its domain, and odd if for all x in its domain.

Even and odd function definition Graph of even function y = x^2 Graph of odd function y = x^3

Examples

Even function: y = x^2 is symmetric about the y-axis.
Odd function: y = x^3 is symmetric about the origin.

Even and odd function examples Even function symmetry Odd function symmetry

Absolute Value and Absolute Value Function

Definition and Properties

The absolute value of a real number x, denoted |x|, is its distance from zero on the number line. It is defined as:

|x| measures distance: |x| is the distance from x to 0; |x - y| is the distance between x and y.

Absolute value as distance Absolute value interval

Absolute Value Function

The absolute value function is defined as . Its domain is all real numbers, and its range is .

Graph of absolute value function

Basic Classes of Functions

Overview

Functions are classified into several types based on their algebraic structure:

Polynomial functions: Linear, quadratic, cubic, etc.
Algebraic functions: Rational, root functions.
Transcendental functions: Trigonometric, exponential, logarithmic.

Classification of basic functions

Linear Functions and Slope

Cartesian Coordinates

The Cartesian coordinate system is used to plot points and graphs in the plane, with axes intersecting at the origin.

Cartesian coordinates

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x between two points:

Slope definition and diagram Slope calculation example Slope of various lines

Equations of Lines

Point-Slope and Slope-Intercept Forms

The equation of a line can be written in several forms:

Point-slope form:
Slope-intercept form:

Point-slope equation Slope-intercept equation

Polynomial Functions

Definition and Types

A polynomial function is any function that can be written as , where n is a non-negative integer and .

Degree: The highest power of x.
Leading coefficient: The coefficient of the highest power.

Polynomial function graphs Graphs of power functions

Algebraic and Transcendental Functions

Algebraic Functions

Algebraic functions include rational and root functions. Rational functions are ratios of polynomials, and root functions involve fractional exponents.

Algebraic function example

Transcendental Functions

Transcendental functions cannot be expressed using only algebraic operations. Examples include trigonometric, exponential, and logarithmic functions.

Transcendental function classification

Piecewise-Defined Functions

Definition and Example

A piecewise-defined function uses different formulas for different parts of its domain. For example:

Piecewise function graph

Transformations of Functions

Shifting, Scaling, and Reflecting

Transformations change the position or shape of a function's graph:

Vertical shift: Add a constant to f(x).
Horizontal shift: Add a constant to x.
Scaling: Multiply f(x) or x by a constant.
Reflection: Multiply f(x) or x by -1.

Vertical shift of quadratic functions Horizontal shift of quadratic functions Transformed absolute value function

Trigonometric Functions

Definition and Properties

Trigonometric functions relate angles to ratios of sides in a right triangle. The six basic functions are sine, cosine, tangent, cosecant, secant, and cotangent.

Trigonometric function definition Trigonometric function quadrants

Summary Table: Function Types and Properties

Type	Definition	Example
Linear	Degree 1 polynomial	y = mx + b
Quadratic	Degree 2 polynomial	y = ax^2 + bx + c
Rational	Ratio of polynomials	y = (x^2 + 1)/(x - 2)
Root	Fractional exponent	y = √x
Trigonometric	Relates angles and sides	y = sin(x)
Exponential	Variable in exponent	y = 2^x
Logarithmic	Inverse of exponential	y = log(x)