Precalculus Textbook

This section begins by revisiting the familiar concept of lines as a specific type of function. It explains that lines can be represented by equations of the form px + qy = r, where p, q, and r are constants. The section illustrates how to determine the slope of a line using the ratio of the rise to the run between two points on the line. It details methods for plotting lines by identifying and plotting points, particularly the x- and y-intercepts (where the line crosses the x and y axes, respectively). The section emphasizes the concept that any two distinct points uniquely define a straight line. It then transitions to a broader discussion of functions, highlighting that functions can be represented via tables of values (input-output pairs), formulas (with explicitly or implicitly defined domains and ranges), and graphs on a coordinate plane. The section also underscores the crucial characteristic of a function: a single input value can only map to one output value; assigning two different outputs to the same input violates the definition of a function. This foundational understanding of lines and functions prepares the reader for the exploration of more complex function types.

The section continues to develop the understanding of functions by describing multiple ways to represent them. It emphasizes the importance of specifying both the domain (the set of all possible input values) and the codomain (a set that contains all possible output values, often taken to be the range if explicitly given). The text provides a clear explanation of the crucial property of functions: a single input must map to only one output. An illustrative example uses a university mentoring program. Matching freshman students with senior mentors defines a function (freshman to mentor), because each freshman has exactly one mentor. However, the reverse assignment (mentor to freshman) does not constitute a function, as one senior could mentor multiple freshmen. The section also highlights that functions can be defined implicitly (such as through a graph) or explicitly (through a given equation, with domains and codomains explicitly stated). Different ways of representing functions such as a table of values, a formula, and a graph are reviewed. The section underscores how the formula and the graphical representation are intrinsically linked. Practical implications and considerations on choosing domains and codomains, depending on whether the range of the function is easily determined or not, are also highlighted.

This subsection elaborates on the graphical representation of functions. It defines the graph of a function f as the set of all points (x, f(x)) on the coordinate plane, where x belongs to the function's domain. It explains how to interpret the graph to determine the output value (y-coordinate) corresponding to a given input value (x-coordinate) by finding the intersection point of the vertical line at x with the graph. This process visually clarifies the concept of input-output mappings within the function's domain and range. The subsection uses examples to illustrate how to extract function values from graphs, including scenarios where the function is partially defined or has limited ranges. The relationship between the algebraic representation of the function (formula) and the graphical representation is repeatedly emphasized. It helps solidify the visual understanding of functions previously introduced in the chapter. It forms the basis for further analysis of functions in the following sections.

This section addresses the problem of finding the roots (or zeros) of polynomial functions. It emphasizes that finding the roots is a crucial step in understanding the behavior of a polynomial. The text highlights that while finding roots of lower-degree polynomials (e.g., quadratics) might be straightforward, finding roots of higher-degree polynomials can be significantly more challenging. The section explains that for rational roots, if a polynomial has a rational root of the form p/q, then p must be a factor of the constant term and q must be a factor of the leading coefficient. It uses examples to illustrate how to identify possible rational roots and subsequently use a calculator to verify them. The importance of utilizing a graphing calculator for finding approximate roots is repeatedly emphasized, especially for higher-degree polynomials where analytical methods can become overly complex. The text also notes that the calculator may only provide approximations, particularly for irrational roots. The section stresses the value of combining algebraic methods with graphical techniques to solve for the roots of polynomials, particularly higher-degree ones.

Building upon the previous discussion of finding roots, this section explores the close relationship between the roots of a polynomial and its factorization. It states that if 'c' is a root of a polynomial f(x), then (x-c) is a factor of f(x). The section demonstrates how to factor a polynomial completely into linear factors once all its roots are known. This process is described as crucial for a thorough understanding of polynomial behavior. The section shows how this factorization can be used to gain insights into the graph of the polynomial, specifically its x-intercepts. The text highlights the utility of this approach, especially when dealing with polynomials of higher degrees. The method relies on the fact that any polynomial of degree n can be fully factored into n linear factors (assuming complex roots are included). Examples of both polynomials with real and complex roots are included to demonstrate the principle and highlight that for polynomials with real coefficients, complex roots always appear in conjugate pairs (a + bi and a - bi).

This section introduces the Fundamental Theorem of Algebra, a cornerstone concept in polynomial theory. The theorem asserts that every polynomial of degree n (with complex coefficients) has exactly n roots (counting multiplicity). The significance of this theorem lies in its guarantee of the existence of roots, even if those roots are complex numbers. The section clarifies that the coefficients of the polynomial can be complex numbers. The importance of this theorem in the context of polynomial factorization is highlighted, stating that once all roots (including complex roots) are identified, a polynomial can be completely factored into linear terms. The text also notes that whenever a polynomial with real coefficients has a complex root (a + bi), its complex conjugate (a - bi) will also be a root. This observation is useful for solving problems involving polynomials where some roots are known, but others are not. It illustrates how the properties of the Fundamental Theorem and the conjugate root relationship allow for deducing the full factorization even if only partial information is initially given.

This section introduces rational functions, defined as the ratio of two polynomial functions. It emphasizes the importance of understanding the behavior of rational functions near their asymptotes and at large values of x. The text explains how to determine the domain of a rational function by identifying values of x that make the denominator zero, these values resulting in vertical asymptotes. It demonstrates how to find the horizontal asymptote by examining the degrees of the numerator and denominator polynomials. For example, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0; if the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients; if the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. The section introduces the concept of a slant asymptote, which occurs when the degree of the numerator is exactly one more than the degree of the denominator. The process of sketching the graph of a rational function is described, emphasizing the consideration of asymptotes and the behavior of the function for large and small values of x. It notes that calculators can have difficulty accurately representing the graph near asymptotes.

This section describes a method for sketching the graphs of rational functions without relying on a calculator. The approach involves analyzing the factored form of the rational function, identifying zeros (roots) and vertical asymptotes. It then explains how to use the signs of the factors in different intervals to determine whether the function is positive or negative in each interval. This information, along with the knowledge of asymptotes, allows for a fairly accurate hand-drawn sketch. The text emphasizes that approximating the graph's behavior for large absolute values of x is helpful in refining the sketch. It also notes the difference in behavior near vertical asymptotes depending on whether the power of the corresponding factor in the denominator is even or odd. If the power is even, the function will approach the asymptote from the same direction on both sides, while an odd power will result in the function approaching the asymptote from opposite directions. This methodical approach allows for a comprehensive understanding of the graph's shape and behavior even without the computational assistance of a graphing calculator. The method described allows the student to draw accurate sketches of rational functions which will help in solving inequalities.

The final part of the section focuses on solving rational inequalities. The text highlights that solving rational inequalities follows a three-step process analogous to solving polynomial inequalities: first solve the corresponding equation, next analyze the intervals created by the solutions obtained in step one by testing sample points, and finally check the endpoints of these intervals. It explains how the graph of the rational function can be employed to identify the intervals where the inequality holds true. This strategy leverages the information obtained from the graphical analysis—the zeros and asymptotes—to determine the solution intervals for the inequality. The section emphasizes the iterative nature of the process and the importance of carefully checking the endpoints to ensure accuracy. It underlines the critical role that understanding the graph of the rational function plays in efficiently solving the inequality, providing a visual and intuitive approach to determining the solution set.

This section introduces the concept of complex numbers, expanding the number system beyond real numbers to include imaginary numbers (multiples of the imaginary unit 'i', where i² = -1). It explains how complex numbers are represented in the form a + bi, where 'a' and 'b' are real numbers, and 'a' is the real part and 'b' is the imaginary part. The text also mentions complex numbers in polar form, suggesting a geometric interpretation that is not elaborated within this section. The importance of complex numbers in the context of polynomial equations is hinted at, setting the stage for the following discussions about the roots of polynomials, and how complex numbers extend the solution set beyond what's possible with real numbers only. The section prepares the reader for the subsequent discussion concerning the roots of polynomials, explaining that some polynomials can only have their full set of roots expressed as complex numbers.

This section directly addresses the topic of complex roots of polynomials. It explains that polynomials can have roots that are complex numbers. It highlights a crucial observation: when a polynomial has real coefficients, its complex roots always occur in conjugate pairs. That is, if (a + bi) is a root, then (a - bi) is also a root. This property is significant in solving polynomial equations, especially when only some of the roots are initially known. The section shows how knowing some of the roots (including the conjugate pairs) enables the complete factorization of the polynomial, emphasizing the connection between roots and factors. The text provides an example of a fourth-degree polynomial where three roots are given (including complex roots), showcasing the methodology of finding the remaining root and factoring the polynomial completely using the roots. This example illustrates the application of the conjugate root property in polynomial factorization.

This section reinforces the connection between the Fundamental Theorem of Algebra and the existence of complex roots of polynomials. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots (counting multiplicity) in the complex number system. This is crucial because it establishes that every polynomial equation has solutions, even if those solutions are complex numbers. The text highlights the theorem's importance in ensuring that every polynomial can be fully factored into linear factors over the complex numbers. It reiterates that for polynomials with real coefficients, complex roots appear in conjugate pairs. The discussion builds upon the previous subsections, consolidating the significance of complex numbers within the context of solving and factoring polynomial equations. The section concludes by linking the Fundamental Theorem to the practical application of finding all roots and expressing polynomials in completely factored form, stressing the completeness of the complex number system in resolving polynomial equations.