Mastering Material Cost Calculations for Rectangular Boxes

Study Notes

Introduction to Functions and Their Representations

A function is a mathematical relationship where one quantity is determined by another.
Functions can be used to analyze various aspects of our environment and daily lives, such as populations, financial markets, and the spread of diseases.
The domain of a function is the set of all allowable inputs, while the range is the set of all possible output values.
An independent variable represents an arbitrary number in the domain, while a dependent variable represents a number in the range.
A function can be visualized as a machine that accepts inputs and produces outputs according to a rule.
The graph of a function consists of input-output pairs plotted as points in the coordinate plane.
If the domain consists of isolated values, the graph is a scatter plot of individual points.
If the input variable can vary continuously through an interval of values, the graph is a curve or line.
The graph of a function can be used to read information, such as the values of the function at specific points.
Functions can be represented verbally, numerically, visually, or algebraically.
Different representations of a function can provide additional insight into the function.
The vertical line test can be used to determine if a graph represents a function.

Introduction to Functions and Piecewise Functions

Functions are mathematical relationships where each input (X) has one unique output (Y).
A function passes the vertical line test, meaning that every vertical line intersects the graph at most once.
Functions can be represented by formulas, tables, or graphs.
The graph of a function can be determined by the vertical line test, which checks if every vertical line intersects the graph at most once.
The graph of a circle centered at the origin with a radius of 5 is not a function because it fails the vertical line test.
The formula for a circle can be solved for Y, resulting in a plus or minus square root, which means it is not a function.
Piecewise functions are functions that have different formulas depending on the value of X.
Absolute value is an example of a piecewise function, where the formula changes depending on whether X is positive or negative.
The absolute value function can be defined as a piecewise function, with different formulas for X greater than or equal to 0 and X less than 0.
Piecewise functions can be graphed by graphing each piece individually, using the appropriate range for each piece.
Piecewise functions allow for different behaviors for different values of X, providing more flexibility in representing mathematical relationships.
Understanding functions and piecewise functions is important in mathematics, as they are fundamental concepts used in various fields of study.Graphing the Function f(x) = x
The text is discussing the process of graphing the function f(x) = x.
The function is in slope-intercept form, with a slope of 1 and a y-intercept of 0.
The graph of f(x) = x crosses the origin (0,0).
The text mentions that a range is given, so only a portion of the graph will be graphed.
The term "domain" is introduced, which refers to the x-values for which the function exists.
The text explains that for this function, the graph exists to the right of the y-axis (x ≥ 0).
The graph of f(x) = x is represented as a straight line with a positive slope of 1.
The text suggests erasing the parts of the graph that do not exist based on the given domain.
The next piece of the graph is discussed, which is f(x) = -x.
The slope of f(x) = -x is in the opposite direction of f(x) = x, going downwards.
Since f(x) = -x is a piece of the graph that has already been graphed, it cannot be erased.
The text ends abruptly, leaving the discussion incomplete.