College Algebra: Functions, Graphs, and Models – Study Notes

Chapter 1: Functions, Graphs, and Models; Linear Functions

Section 1.1: Functions and Models

This section introduces the foundational concept of functions in algebra, their representations, and how they are used to model real-world phenomena. Students will learn to identify functions, determine domains and ranges, use function notation, and apply mathematical models to practical situations.

Objectives

• Determine if a table, graph, or equation defines a function.

• Find the domains and ranges of functions.

• Create a scatter plot of a set of ordered pairs.

• Use function notation to evaluate functions.

• Apply real-world information using a mathematical model.

Definition of a Function

A function is a rule or correspondence that assigns to each element of one set (called the domain) exactly one element of a second set (called the range).

• A function may be defined by a set of ordered pairs, a table, a graph, an equation, or a verbal description.

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

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

Example: Body Temperature Conversion

To convert a temperature from Celsius to Fahrenheit, use the following formula:

$F = \frac{9}{5}C + 32$

• If a child's temperature is 37°C, then:

• $F = \frac{9}{5}(37) + 32 = 98.6$

• This indicates that the child's temperature is normal.

Domains and Ranges

The domain and range of a function can be determined from a graph, table, or set of ordered pairs.

Example: Table and Graph of Ordered Pairs

x	y
-5	-7
-3	-2
-1	6
0	-2
3	5
5	-7

• Domain: {-5, -3, -1, 0, 3, 5}

• Range: {-7, -2, 6, 5}

• Note: Each y-value is listed only once in the range.

Example: Domain and Range from a Graph

• For a graph with $-3 \leq x \leq 3$, the domain is $[-3, 3]$.

• If $-27 \leq y \leq 39$, the range is $[-27, 39]$.

Example: Infinite Domain or Range

• If a graph continues indefinitely to the left, the domain may be $(-\infty, 4]$.

• If the graph continues upward, the range may be $[0, \infty)$.

Recognizing Functions

To determine whether a relationship is a function, check if each input corresponds to exactly one output.

• If a table lists each year with only one value for the number of physicians, it represents a function.

• Domain: The set of years.

• Range: The set of physician counts.

Function Notation

Functions are often written using notation such as $f(x)$, which denotes the output of function $f$ for input $x$.

• Example: $f(x) = 2x^3 + 5x^2 - 28x - 15$

• To find $f(2)$, substitute $x = 2$:

• $f(2) = 2(8) + 5(4) - 28(2) - 15 = 16 + 20 - 56 - 15 = -35$

• To find $f(4)$, substitute $x = 4$:

• $f(4) = 2(64) + 5(16) - 28(4) - 15 = 128 + 80 - 112 - 15 = 81$

Vertical Line Test

The vertical line test is a graphical method to determine if a curve is the graph of a function.

• If any vertical line intersects the graph at more than one point, the graph does not represent a function.

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

Applications of Functions

Functions are used to model real-world scenarios, such as population growth, market value, or workforce statistics.

Example: Medical 3D Printing Market

• The value of the medical 3D printing market $V(t)$ (in millions of dollars) is modeled by:

• $V(t) = 3.77t^2 + 24.4t + 319.9$

• Here, $t$ is the number of years after 2010.

• To find the market value in 2025, set $t = 15$:

• $V(15) = 3.77(225) + 24.4(15) + 319.9 = 848.25 + 366 + 319.9 = 1534.15$

• So, the projected market in 2025 is approximately $1,534$ million or $1.534$ billion.

Example: Workforce Statistics

• If $g(2020) = 21$, this means there were 21 million older men in the workforce in 2020.

• If $g(2030) = 19.6$, then 19.6 million older men are projected for 2030.

• The maximum number during the period shown may be at a specific year, e.g., 21 million in 2020.

Summary Table: Ways to Represent Functions

Representation	Description	Example
Ordered Pairs	List of (x, y) values	{(-5, -7), (-3, -2), ...}
Table	Tabular format of x and y	See above table
Graph	Plot on coordinate axes	Scatter plot or curve
Equation	Algebraic rule	$y = 2x^3 + 5x^2 - 28x - 15$
Verbal Description	Words describing the relationship	"The number of physicians as a function of year"

Additional info: Some examples and equations have been expanded for clarity and completeness. The market value formula for medical 3D printing was inferred from context and typical quadratic modeling in applications.