Functions and Models: Foundations of College Algebra

Functions, Graphs, and Models

Definition of a Function

A function is a mathematical relationship that assigns to each element in a set called the domain (input) exactly one element in a second set called the range (output). Functions can be represented in several ways, including tables, graphs, equations, and sets of ordered pairs.

• Domain: The set of all possible input values.

• Range: The set of all possible output values.

• Ordered pairs: Each input is paired with exactly one output.

Representations of Functions

Functions can be described using:

• Tables: Lists of input-output pairs.

• Graphs: Visual representations showing how outputs change with inputs.

• Equations: Algebraic expressions relating input and output.

• Data: Sets of measured or observed values.

Example: Temperature Conversion

The relationship between Celsius and Fahrenheit temperatures is a function. It can be represented as a table, a set of ordered pairs, or an equation.

Celsius Degrees (°C)	Fahrenheit Degrees (°F)
-20	-4
-10	14
-5	23
0	32
25	77
50	122
100	212

The conversion formula is: $F = \frac{9}{5}C + 32$

Example: Blood Alcohol Percent

Each alcoholic drink increases the blood alcohol percent of a 90-pound woman by 0.05. This relationship is a function, and can be modeled by the equation $y = 0.05x$, where $x$ is the number of drinks and $y$ is the blood alcohol percent.

Number of Drinks	Blood Alcohol Percent
0	0.00
1	0.05
2	0.10
3	0.15
4	0.20
5	0.25
6	0.30
7	0.35
8	0.40
9	0.45

Function Notation and Evaluation

Function Notation

Functions are often written using notation such as $f(x)$, which means "the output of function f for input x." To evaluate a function, substitute the input value into the equation.

• Example: If $f(x) = 2x^3 + 5x^2 - 28x - 15$, then $f(-2)$ and $f(4)$ can be found by substituting $x = -2$ and $x = 4$ into the equation.

Functions as Models

Using Functions to Model Data

Functions can be used to model real-world data, such as the relationship between the number of drinks and blood alcohol percent, or between Celsius and Fahrenheit temperatures. A function is a model if it accurately represents the data.

• Example: The function $y = 0.05x$ models the blood alcohol percent for a 90-pound woman.

• Example: The function $F = \frac{9}{5}C + 32$ models the conversion between Celsius and Fahrenheit.

Applications of Functions

Financial Models: Mortgage Payments

Functions can describe financial relationships, such as the number of years required to pay off a mortgage based on the loan amount. The table below shows the relationship between the mortgage amount $A$ and the number of years $t$ to pay off the loan at a fixed monthly payment and interest rate.

Amount A ($)	t (years)
40,000	5
69,000	10
89,000	15
103,000	20
120,000	30

• Example: If a couple finances $103,000, they will make payments for 20 years.

• Function notation: $t = f(A)$ where $A$ is the mortgage amount.

Summary Table: Types of Function Representations

Representation	Description	Example
Table	Lists input-output pairs	Blood alcohol percent table
Graph	Visual plot of function	Graph of $f(x)$
Equation	Algebraic formula	$f(x) = 2x^3 + 5x^2 - 28x - 15$
Ordered pairs	Set of (input, output) pairs	{(-20, -4), (0, 32), (100, 212)}

Key Concepts

• Functions describe relationships between inputs and outputs.

• Functions can be represented in multiple ways: tables, graphs, equations, and ordered pairs.

• Function notation ($f(x)$) is used to evaluate and describe functions.

• Functions are used to model real-world data and solve practical problems.

Examples and Applications

• Temperature conversion: Use the formula $F = \frac{9}{5}C + 32$ to convert Celsius to Fahrenheit.

• Blood alcohol percent: Use $y = 0.05x$ to estimate blood alcohol percent based on number of drinks.

• Mortgage payments: Use tables and function notation to determine payment periods for different loan amounts.

Additional info: Academic context was added to clarify the definition of functions, function notation, and the use of functions as models in real-world applications.