Functions and Relations

Ordered Pairs, Domain, and Range

In mathematics, a relation is a set of ordered pairs. The domain of a relation is the set of all possible input values (usually x-values), and the range is the set of all possible output values (usually y-values).

• Domain: The set of all first elements in the ordered pairs.

• Range: The set of all second elements in the ordered pairs.

• Function: A relation is a function if each input (x-value) corresponds to exactly one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Example: For the relation {(2,3), (1,0), (4,-5), (2,-3)}, the domain is {1,2,4} and the range is {0,3,-5,-3}. This is not a function because the input 2 corresponds to two different outputs (3 and -3).

Graphing Functions and Determining Domain and Range

Graphs can be used to visually determine the domain and range of a function.

• Domain: The set of all x-values for which the function is defined.

• Range: The set of all y-values that the function attains.

• Use the vertical line test to check if a graph represents a function.

Example: For a parabola opening downward with vertex at (0,7), the domain is and the range is .

Evaluating Functions and Table of Values

Function Evaluation

To evaluate a function, substitute the given value into the function's formula.

• Example: Given , evaluate :

Table of Values

A table of values lists input-output pairs for a function, which can be used to plot the graph.

Linear Equations and Slope

Slope Calculation

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.

• Formula:

• Horizontal lines: Slope

• Vertical lines: Slope is undefined

Example: For points (5,3) and (1,4):

Point-Slope and Slope-Intercept Forms

Linear equations can be written in different forms:

• Slope-intercept form:

• Point-slope form:

Example: Find the equation of a line passing through (2,4) with slope :

• Point-slope form:

• Simplified:

Parallel and Perpendicular Lines

• Parallel lines: Have the same slope.

• Perpendicular lines: Slopes are negative reciprocals.

Example: A line perpendicular to has slope .

Finding x- and y-intercepts

The x-intercept is where the graph crosses the x-axis (), and the y-intercept is where it crosses the y-axis ().

• Set and solve for to find the x-intercept.

• Set and solve for to find the y-intercept.

Example: For :

• x-intercept:

• y-intercept:

Linear Models and Applications

Interpreting Linear Models

Linear models are used to describe relationships between variables, such as sales over time.

• General form:

• Slope (m): Rate of change

• y-intercept (b): Initial value

Example: models appliances sold, where 72 is the rate per year and 125 is the initial number sold.

Using Tables to Model Data

Tables can be used to find linear relationships and make predictions.

• Find the slope:

• Linear model: (where x is years since 2015)

Applications: Price Prediction

Linear models can be used to predict future values, such as the price of milk over time.

• Given , where t is years since 2005, predict the price in 2021:

Example: In 2005, the price of a gallon of milk was $2.83. In 2021, it is predicted to be $6.51.

Summary Table: Key Concepts

Additional info: These notes expand on the original handwritten content by providing full definitions, formulas, and context for each concept, ensuring a self-contained study guide suitable for College Algebra exam preparation.