Library of Functions

Overview

The library of functions consists of basic functions that serve as building blocks for more complex mathematical models. Understanding their properties and graphs is essential for precalculus students.

Reciprocal Function

The reciprocal function is defined as . It is important for its unique graph and properties.

• Domain: All real numbers except

• Range: All real numbers except

• x-intercept: None

• y-intercept: None

• Symmetry: Symmetrical with respect to the origin (odd function)

• Increasing/Decreasing: Decreasing on and

• Local/Absolute Extrema: None

Example: ,

Square Function

The square function is defined as . It is a fundamental quadratic function.

• Domain: All real numbers

• Range:

• x-intercept:

• y-intercept:

• Symmetry: Symmetrical about the y-axis (even function)

• Increasing: On

• Decreasing: On

• Local/Absolute Minimum: At

• Local/Absolute Maximum: None

Example: ,

Cube Function

The cube function is defined as . It is another basic polynomial function.

• Domain: All real numbers

• Range: All real numbers

• x-intercept:

• y-intercept:

• Symmetry: Symmetrical with respect to the origin (odd function)

• Increasing: Everywhere

• Decreasing: Nowhere

• Local/Absolute Extrema: None

Example: ,

Constant Function

The constant function is defined as , where is a real number. Its graph is a horizontal line.

• Domain: All real numbers

• Range:

• x-intercept: None (unless )

• y-intercept:

• Symmetry: Even function

• Increasing/Decreasing: Neither

• Local/Absolute Extrema: All points are both maximum and minimum

Example:

Identity Function

The identity function is defined as . Its graph is a straight line through the origin.

• Domain: All real numbers

• Range: All real numbers

• x-intercept:

• y-intercept:

• Symmetry: Symmetrical with respect to the origin (odd function)

• Increasing: Everywhere

• Decreasing: Nowhere

• Local/Absolute Extrema: None

Example:

Greatest Integer Function

The greatest integer function (also known as the floor function) is defined as , which returns the greatest integer less than or equal to .

• Domain: All real numbers

• Range: All integers

• Graph: Step-like appearance

Example: ,

Piecewise-Defined Functions

Definition and Properties

A piecewise-defined function uses different formulas for different parts of its domain. These functions are useful for modeling situations where a rule changes based on input value.

• Notation:

• Graph: May have jumps, holes, or different shapes in different intervals

Example: Cellular Phone Plan

Consider a function for monthly cost based on minutes used:

• For ,

• For ,

• For ,

Example: Piecewise Quadratic Function

Consider defined as:

• for

• for

Graphing Piecewise Functions

To graph a piecewise function, plot each segment according to its formula and domain. Pay attention to open and closed endpoints.

Summary Table: Library Functions

Applications

Piecewise Functions in Real Life

Piecewise functions are used to model real-world situations such as pricing plans, tax brackets, and physical phenomena where rules change at certain thresholds.

Example: A phone plan with a flat rate up to a certain number of minutes, then a per-minute charge after that, is modeled by a piecewise function.

Key Takeaways

• Mastery of library functions is foundational for precalculus.

• Piecewise-defined functions require careful attention to domain and formula for each segment.

• Graphing these functions involves plotting each segment and marking endpoints correctly.