Functions, Domain, and Range: Foundations of College Algebra

Functions and Their Properties

Definition of a Function

A function is a relation that assigns to each element in a set called the domain exactly one element in a set called the range. In other words, a function pairs every input value with one and only one output value.

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

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

Example: For the function $f(x) = x^2$, the domain is all real numbers, and the range is all real numbers greater than or equal to zero.

Input and Output

• The input of a function is the value you substitute into the function (the independent variable, usually x).

• The output is the result after applying the function rule to the input (the dependent variable, usually y or f(x)).

Example: If $f(x) = 2x + 3$ and the input is $x = 4$, then the output is $f(4) = 2(4) + 3 = 11$.

Identifying Domain and Range from Data

Tables and Real-World Contexts

When given a table of values or real-world data, the domain is typically the set of all input values (such as years, time, or other independent variables), and the range is the set of all output values (such as sales, population, or other dependent variables).

• Example: In a table showing U.S. sales of personal computers by year, the year is the domain, and the sales is the range.

• Sometimes, the roles of input and output can be reversed depending on the context or the question being asked.

Graphs and Functions

Graphical Representation of Functions

A graph is a function if and only if every vertical line drawn through the graph intersects it at most once. This is known as the Vertical Line Test.

• If a graph passes the vertical line test, it represents a function.

• If any vertical line crosses the graph more than once, the graph does not represent a function.

Example: The graph of $y = x^2$ passes the vertical line test and is a function. The graph of a circle, $x^2 + y^2 = 1$, does not pass the test and is not a function.

Domain of Specific Functions

Finding the Domain

• For rational functions such as $y = \frac{x^2}{x-5}$, the domain is all real numbers except where the denominator is zero.

• For radical functions such as $y = \sqrt{x+5}$, the domain is all real numbers for which the expression under the square root is non-negative.

Examples:

• Domain of $y = \frac{x^2}{x-5}$: All real numbers except $x = 5$.

• Domain of $y = \sqrt{x+5}$: All real numbers $x$ such that $x + 5 \geq 0$, or $x \geq -5$.

Scatter Plots and Mathematical Models

Scatter Plots

A scatter plot is a graphical representation of data points on a coordinate plane. Each point represents a pair of values (input, output) from a data set.

• Scatter plots are useful for visualizing relationships between two variables.

• They can help identify trends, patterns, or correlations in data.

Mathematical Models

A mathematical model is an equation or function that represents real-world phenomena. Models are used to describe, predict, or analyze relationships between variables.

• Common types of models include linear, quadratic, exponential, and logarithmic functions.

• Models are constructed based on observed data and are used to make predictions or understand underlying patterns.

Example: A linear model for predicting sales based on years might be $S = 5000 + 200t$, where $t$ is the number of years after a starting point.

Classifying Relations as Functions

Determining if a Relation is a Function

• A relation is a function if each input value corresponds to exactly one output value.

• To determine if a table, graph, or equation represents a function, check that no input value is paired with more than one output value.

Example: The set of pairs {(1,2), (2,3), (3,4)} is a function. The set {(1,2), (1,3), (2,4)} is not a function because the input 1 is paired with two different outputs.

Aligning Data and Interpreting Variables

Aligning Data in Tables and Graphs

• When working with data, ensure that each input value is correctly matched with its corresponding output value.

• Pay attention to the definitions of variables, especially when interpreting real-world problems (e.g., what does the variable t represent?).

Example: If t represents the number of years after 2000, then t = 25 corresponds to the year 2025.

Summary Table: Function Properties

Type of Function	General Form	Domain	Range	Example
Linear	$y = mx + b$	All real numbers	All real numbers	$y = 2x + 3$
Quadratic	$y = ax^2 + bx + c$	All real numbers	$y \geq$ minimum (if $a > 0$)	$y = x^2$
Rational	$y = \frac{p(x)}{q(x)}$	All real numbers except where $q(x) = 0$	Varies	$y = \frac{x^2}{x-5}$
Radical	$y = \sqrt{x + a}$	$x + a \geq 0$	$y \geq 0$	$y = \sqrt{x+5}$

Additional info: These notes expand on the worksheet prompts by providing definitions, examples, and a summary table for common function types, as well as context for interpreting domain and range in real-world and mathematical settings.