Library of Functions and Piecewise-Defined Functions

Section 1.4 Library of Functions; Piecewise-Defined Functions

Introduction

This section introduces the fundamental functions commonly used in precalculus, known as the "library of functions," and explores the concept and analysis of piecewise-defined functions. Understanding these basic functions and their properties is essential for graphing and analyzing more complex mathematical relationships.

Library of Functions

Constant and Identity Functions

• Constant Function: Defined as $f(x) = b$, where $b$ is a real number. The graph is a horizontal line.

• Identity Function: Defined as $f(x) = x$. The graph is a straight line passing through the origin with a slope of 1.

Square and Cube Functions

• Square Function: $f(x) = x^2$

  • Domain: All real numbers

  • Range: $[0, \infty)$

  • Even function (symmetric with respect to the y-axis)

  • Absolute minimum at $x = 0$

• Cube Function: $f(x) = x^3$

  • Domain and Range: All real numbers

  • Odd function (symmetric with respect to the origin)

  • No local minima or maxima

Square Root and Cube Root Functions

• Square Root Function: $f(x) = \sqrt{x}$

  • Domain: $[0, \infty)$

  • Range: $[0, \infty)$

  • Neither even nor odd

  • Absolute minimum at $x = 0$

• Cube Root Function: $f(x) = \sqrt[3]{x}$

  • Domain and Range: All real numbers

  • Odd function (symmetric with respect to the origin)

  • Increasing everywhere

Reciprocal and Absolute Value Functions

• Reciprocal Function: $f(x) = \frac{1}{x}$

  • Domain: $x \neq 0$

  • Range: $y \neq 0$

  • Odd function (symmetric with respect to the origin)

  • Vertical and horizontal asymptotes at $x = 0$ and $y = 0$

• Absolute Value Function: $f(x) = |x|$

  • Domain: All real numbers

  • Range: $[0, \infty)$

  • Even function (symmetric with respect to the y-axis)

  • Absolute minimum at $x = 0$

Greatest Integer Function

• Greatest Integer Function (Floor Function): $f(x) = \text{int}(x)$ or $f(x) = \lfloor x \rfloor$

  • Returns the greatest integer less than or equal to $x$

  • Graph is a step function

Graphing and Analyzing Library Functions

Example: Cube Root Function

• Function: $f(x) = \sqrt[3]{x}$

• Symmetry: Odd function (symmetric with respect to the origin)

• Intercepts: Both x- and y-intercepts at (0, 0)

• Table of Values:

• Graph:

Example: Absolute Value Function

• Function: $f(x) = |x|$

• Symmetry: Even function (symmetric with respect to the y-axis)

• Intercepts: Both x- and y-intercepts at (0, 0)

• Table of Values:

• Graph:

Piecewise-Defined Functions

Definition and Analysis

A piecewise-defined function is a function that is defined by different expressions for different intervals of the domain. Each "piece" applies to a specific part of the domain.

• To analyze a piecewise-defined function:

  • Identify the intervals and the corresponding expressions.

  • Find the domain by considering all intervals.

  • Find intercepts by solving for $f(x) = 0$ and $f(0)$ in each piece.

  • Graph each piece over its interval, paying attention to endpoints (open or closed circles).

  • Determine the range from the graph.

Example: Graphing a Piecewise-Defined Function

• Given a function $f$ defined by different expressions on different intervals, graph each piece and combine them to form the complete graph.

Example: Analyzing a Piecewise-Defined Function

• Domain: Determined by the union of the intervals for which each piece is defined.

• Intercepts: Found by evaluating $f(0)$ for the y-intercept and solving $f(x) = 0$ for x-intercepts in each piece.

• Graph: Draw each piece over its interval, noting open or closed endpoints.

• Range: Determined from the graph as the set of all possible output values.

Applications of Piecewise-Defined Functions

Example: Cost of Electricity

• Electricity cost is modeled as a piecewise-defined function, where the rate changes after a certain usage threshold.

• For $x$ kWh used:

  • If $0 \leq x \leq 700$, $C = 20 + 0.1148x$

  • If $x > 700$, $C = 20 + 0.1148 \times 700 + 0.1160(x - 700)$

• The graph consists of two linear pieces with different slopes, meeting at the point $(700, 100.36)$.

Additional info: Piecewise-defined functions are commonly used to model real-world situations where a rule or relationship changes at certain thresholds, such as tax brackets, shipping rates, or utility pricing.