In particular, it is expected that students will come to this course with a strong conceptual foundation in ratios, rates, and proportional relationships and an understanding of simple linear and non-linear patterns of growth and their representation in the coordinate plane. Facility with real number operations and integer exponents and roots is assumed. Students entering Algebra I with this set of knowledge and skills should be prepared to extend and deepen their understanding of numbers and their application. Algebra I has a strong emphasis on extending and formalizing understanding of advanced linear as well as simple exponential and quadratic functions and equations. To support this work, students in Algebra I will be expected to acquire more advanced number skills including facility with rational exponents and radical expressions.

This course is the first in a three-year traditional course sequence that has been enhanced by the addition of topics taken from data analysis, statistics, and discrete mathematics. The algebraic reasoning and skills developed in this course will be applied to geometric contexts in the second traditional course and will form the foundation for the work done in the third course where more advanced functions are introduced and generalizations across all function families more fully explored.

Appropriate use of technology is expected in all work in this Traditional Plus course sequence. In Algebra I, this includes employing technological tools to assist students in the formation and testing of conjectures, creating graphs and data displays, and determining and assessing lines of fit for data. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of Algebra I to those encountered in middle school mathematics as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for the Algebra I course. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Applications of Numbers
• Mathematical Vocabulary and Logic
• Properties of Irrational and Real Numbers
• Patterns of Growth Through Iteration
• Linear, Proportional, and Piecewise-Linear Functions
• Linear Equations, Inequalities, and Systems of Linear Equations
• Exponents and Simple Exponential Functions
• Quadratic Functions and Equations Over the Real Numbers

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this Algebra I course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for Algebra I.

• Real Numbers Operations
• Whole Number Exponents and Roots
• Ratios, Rates, and Proportions
• Functions and Graphing in the Coordinate Plane
• Linear Equations and Inequalities and Their Applications
• Linear and Simple Exponential Patterns of Growth
• Elementary Data Analysis

A. Applications of Numbers

This course begins by reinforcing student understanding of real numbers and their multiple representations. Measurement situations, including those that employ derived measures and estimation, provide opportunities for students to apply real numbers. Scientific and factorial notation extends students’ repertoire of mathematical representations to facilitate the communication of the extremely large and small numbers that often arise in applications.

Successful students will:

A1 Extend and apply understanding about rates and ratios, estimation, and measurement to derived measures including weighted averages; use appropriate units and unit analysis to express and check solutions.

Derived measures are those achieved through calculations with measurements that can be taken directly.

1. Identify applications that can be expressed using derived measures or weighted averages.

   Examples: Percent change and density are examples of derived measures; grade averages, stock market indexes, the consumer price index, and unemployment rates are examples of weighted averages.

A2 Interpret, compare, and use extreme numbers involving significant figures, orders of magnitude, scientific notation; determine a reasonable degree of precision when making calculations or estimations.

1. Identify applications that involve extreme numbers.

   Examples: Extreme numbers include lottery odds, national debt, astronomical distances, or the size of a connector in a microchip.

2. Assess the amount of error resulting from estimation and determine whether the error is within acceptable tolerance limits.

A3 Determine the number of ways events can occur using permutations, combinations, and other systematic counting methods.

A permutation is a rearrangement of distinct items in which their order matters; a combination is a selection of a given number of distinct items from a larger number without regard to their arrangement (i.e., in which their order does not matter).

1. Know and apply organized counting techniques such as the Fundamental Counting Principle.

   The Fundamental Counting Principal is a way of determining the number of ways a sequence of events can take place. If there are n ways of choosing one thing and m ways of choosing a second after the first has been chosen, then the Fundamental Counting Principal says that the total number of choice patterns is
   n · m.

   Examples: How many different license plates can be formed with two letters and three numerals? If the letters had to come first, how many letters would be needed to create at least as many different license plate numbers? How many different subsets are possible for a set with six elements?

2. Distinguish between counting situations that do not permit replacement and situations that do permit replacement.

   Examples: How many different four-digit numbers can be formed if the first digit must be non-zero and each digit may be used only once? How many are possible if the first digit must be non-zero but digits can be used any number of times?

3. Distinguish between situations where order matters and situations where it does not; select and apply appropriate means of computing the number of possible arrangements of the items in each case.

A4 Interpret and simplify expressions involving factorial notation; recognize simple cases where factorial notation may be used to express a result.

Examples: Interpret 6! as the product 6 · 5 · 4 · 3 · 2 · 1; recognize that = 15 · 14 · 13 = 2,730; express the number of possible orders in which 7 names can be listed on a ballot as 7!

B. Mathematical Vocabulary and Logic

The deductive reasoning that leads from assumptions and definitions to an irrefutable conclusion sets mathematics apart from the inductive reasoning found in science or the situational reasoning found in most other disciplines. The grounding for such logical thought and expression is a solid conceptual understanding of sets and their operations—also important as the theoretical foundation for digital systems and for describing logic circuits and formulating search engine queries—knowledge and careful application of mathematical vocabulary and notation and attention to mathematical syntax. The mathematical content encountered earlier in middle school courses as well as common media outlets provide excellent resources through which students can learn to carefully analyze and interpret statements, explore generalizations, and formulate, test, critique, and verify or refute conjectures.

Successful students will:

B1 Know the concepts of sets, elements, empty set, relations (e.g., belong to), and subsets and use them to represent relationships among objects and sets of objects.

1. Recognize and use different methods to define sets (lists, defining property).
2. Perform operations on sets: union, intersection, complement.

   Example: Use Boolean search techniques to refine online bibliographic searches.

3. Create and interpret Venn diagrams to solve problems.
4. Identify whether a given set is finite or infinite; give examples of both finite and infinite sets.

B2 Use and interpret mathematical notation, terminology, and syntax.

1. Use and interpret common mathematical terminology.

   Examples: Use and interpret conjunctions, disjunctions, and negations ("and," "or," "not"); use and interpret terms of causation ("if...then") and equivalence ("if and only if").

2. Describe logical statements using terms such as assumption, hypothesis, conclusion, converse, and contraposition.
3. Recognize uses of logical terms in everyday language, noting the similarities and differences between common use and their use in mathematics.

B3 Analyze and apply logical reasoning.

1. Distinguish between deductive and inductive reasoning; identify the strengths of each type of reasoning and its application in mathematics.

   Inductive reasoning is based on observed patterns and can be used in mathematics to generate conjectures after which deductive reasoning can be used to show that the conjectures are true in all circumstances. Inductive reasoning cannot prove propositions; valid conclusions and proof require deduction.

2. Explain and illustrate the importance of generalization in mathematics and its relationship to inductive and deductive reasoning.
3. Make, test, and confirm or refute conjectures using a variety of methods; construct simple logical arguments and proofs; determine simple counterexamples.
4. Recognize syllogisms, tautologies, and circular reasoning and use them to assess the validity of an argument.
5. Recognize and identify flaws or gaps in the reasoning used to support an argument.

   Example: Recognize that A → B does not imply that B → A.

6. Explain reasoning in both oral and written forms.

B4 Analyze and apply algorithms for searching, for sorting, and for solving optimization problems.

1. Identify and apply algorithms for searching, such as sequential and binary.
2. Describe and compare simple algorithms for sorting, such as bubble sort, quick sort, and bin sort.

   Example: Compare strategies for alphabetizing a long list of words; describe a process for systematically solving the Tower of Hanoi problem.

3. Know and apply simple optimization algorithms.

   Example: Use a vertex-edge graph (network diagram) to determine the shortest path needed to accomplish some task.

C. Properties of Irrational and Real Numbers

Building on the understanding of whole number exponents, simple roots, and the relationship between the square root and the exponent ½, students will develop an understanding of the impact of a negative exponent and generalize the properties of exponents to all rational exponents. Facility with both exponential and radical expression of numbers opens up the ability to compute in either form and to deepen student insight into the real number system. Properties such as the density of the rational numbers can be explored on an informal level, through example and explanation rather than formal proof, and lead naturally to the ability to estimate irrational numbers to any given degree of precision.

Successful students will:

C1 Interpret and apply integral and rational exponents in numerical expressions.

1. Use rational exponents to rewrite numerical expressions.

   Examples:

2. Convert between forms of numerical expressions involving roots; perform operations on numbers expressed in exponential or radical form.

   Examples: Convert and use the understanding of this conversion to perform similar calculations and to compute with numbers in radical form; rewrite as a fraction having only positive exponents and as a fraction in lowest terms.

C2 Establish simple facts about rational and irrational numbers using logical arguments and examples.

Examples: Explain why, if r and s are rational, then both r + s and rs are rational. For example, both ¾ and 2.3 are rational in , which is the ratio of two integers, hence rational. Give examples to show that, if r and s are irrational, then r + s and rs could be either rational or irrational (for example, is irrational whereas is rational). Show that a given interval on the real number line, no matter how small, contains both rational and irrational numbers.

C3 Given a degree of precision, determine a rational approximation to that degree of precision for an irrational number expressed using rational exponents or radicals.

D. Patterns of Growth Through Iteration

This is an opportunity to deepen and extend understanding of linear and simple exponential patterns of growth studied in middle school. Sequences can be represented on a coordinate plane and the characteristics of the resulting graphs tied to previously encountered concepts, such as slope.

Successful students will:

D1 Analyze, interpret, and describe relationships represented iteratively and recursively, including those produced using a spreadsheet.

Examples: Recognize that the sequence defined by "First term = 5; each term after the first is six more than the preceding term" is the sequence whose first seven terms are 5, 11, 17, 23, 29, 35, and 41; recognize that the result of repeatedly squaring a number between –1 and 1 appears to approach zero, while the result of repeatedly squaring a number less than –1 or greater than 1 appears to continue to increase; determine empirically how many steps are needed to produce four-digit accuracy in square roots by iterating the operations divide and average.

D2 Generate and describe sequences having specific characteristics; use calculators and spreadsheets effectively to extend sequences beyond a relatively small number of terms.

1. Generate and describe the factorial function or the Fibonacci sequence recursively.
2. Generate and describe arithmetic sequences recursively; identify arithmetic sequences expressed recursively.

   Arithmetic sequences are those in which each term differs from its preceding term by a constant difference. To describe an arithmetic sequence, both the starting term and the constant difference must be specified.

   Example: a₁ = 5, a_n+1 = a_n + 2 describes the arithmetic sequence 5, 7, 9, 11,...

3. Generate and describe geometric sequences recursively; identify geometric sequences expressed recursively.

   Geometric sequences are those in which each term is a constant multiple of the term that precedes it. To describe a geometric sequence both the starting term and the constant multiplier (often called the common ratio) must be specified.

   Example: a₁ = 3, a_n+1 = 2a_n describes the geometric sequence 3, -6, 12, -24,...

4. Given an irrational number expressed using rational exponents or radicals, find increasing and decreasing sequences that converge to that number and show that the first terms of these sequences satisfy the right inequalities.

   Example: 1 < 1.4 < 1.41 < 1.414 < ... < < 1.415 < 1.42 < 1.5 < 2 since 1² = 1 < (1.4)² = 1.96 < (1.41)² = 1.9881 < (1.414)² = 1.999396 < ... < = 2 < ... < (1.415)² = 2.002225 < (1.42)² = 2.0164 < (1.5)² = 2.25 < 2² = 4

D3 Demonstrate the effect of compound interest, exponential decay, or exponential growth using iteration.

Examples: Using a spreadsheet, enter the amount of a loan, the monthly interest rate, and the monthly payment in a spreadsheet. The formula (loan amount) · (1 + monthly interest rate) - (monthly payment) gives the amount remaining monthly on the loan at the end of the first month, and the iterative "fill down" command will show the amount remaining on the loan at the end of each successive month; a similar process using past data about the yearly percent increase of college tuition and annual inflation rate will provide an estimate of the cost of college for a newborn in current dollar equivalents.

1. Identify the diminishing effect of increasing the number of times per year that interest is compounded and relate this to the notion of instantaneous compounding.

E. Linear, Proportional, and Piecewise-Linear Functions

Linear patterns of growth are a focus of the middle school curriculum. Description, analysis, and interpretation of lines should continue to be reinforced and extended as students work with linear functions. The reciprocal functions introduced here should be linked back to student experience with proportions and with the simple exponential patterns of growth studied in middle school. Absolute value, step, and other piecewise-linear functions are included to extend student facility with linear growth to situations that are defined differently over subsets of their domains, a fairly common phenomenon in real-life contexts. Also included here is a first look at how changes in parameters affect the graph of a function.

Successful students will:

E1 Recognize, graph, and use direct proportional relationships.

A proportion is composed of two pairs of real numbers, (a, b) and (c, d), with at least one member of each pair non-zero, such that both pairs represent the same ratio. A linear function in which f(0) = 0 represents a direct proportional relationship. The function f(x) = kx, where k is constant, describes a direct proportional relationship.

1. Analyze the graph of direct proportional relationships, f(x) = kx, and identify its key characteristics.

   A direct proportional relationship is represented by a line that passes through the origin (0, 0) whose slope is the constant of proportionality.

2. Compare and contrast the graphs of x = k, y = k, and y = kx, where k is a constant.
3. Recognize and provide a logical argument that if f(x) is a linear function, g(x) = f(x) - f(0) represents a direct proportional relationship.

   Example: If f(x) is linear, f(x) = mx + b and f(0) = 0 + b, so g(x) = f(x) - f(0) = mx + b - b = mx. This means that g(0) = 0, so the function y = g(x) is a direct proportional relationship.

4. Recognize quantities that are directly proportional and express their relationship symbolically.

   Example: The relationship between length of the side of a square and its perimeter is directly proportional.

E2 Recognize, graph, and use reciprocal relationships.

A function of the form f(x) = k/x where k is constant describes a reciprocal relationship. The term “inversely proportional” is sometimes used to identify such relationships, however, this term can be very confusing since the word "inverse" is also used in the term "inverse function" (the function y = f^-1(x) with the property that f f^-1(x) = f^-1 f(x) = x, which describes the identity function).

1. Analyze the graph of reciprocal relationships, f(x) = k/x, and identify its key characteristics.

   The graph of f(x) = k/x is not a straight line and does not cross either the x– or the y–axis (i.e., there is no value of x for which f(x) = 0, nor is there any value for f(x) if x= 0). It is a curve consisting of two disconnected branches, called a hyperbola.

2. Recognize quantities that are inversely proportional and express their relationship symbolically.

   Example: The relationship between lengths of the base and side of a rectangle with fixed area is inversely proportional.

E3 Distinguish among and apply linear, direct proportional, and reciprocal relationships; identify and distinguish among applications that can be expressed using these relationships.

1. Identify whether a table, graph, formula, or context suggests a linear, direct proportional, or reciprocal relationship.
2. Create graphs of linear, direct proportional, and reciprocal functions by hand and using technology.
3. Distinguish practical situations that can be represented by linear, direct proportional, or reciprocal relationships; analyze and use the characteristics of these relationships to answer questions about the situation.

E4 Explain and illustrate the effect of varying the parameters m and b in the family of linear functions, f(x) = mx + b, and varying the parameter k in the families of direct proportional and reciprocal functions, f(x) = kx and f(x) = k/x, respectively.

E5 Identify key characteristics of absolute value, step, and other piecewise-linear functions and graph them.

1. Interpret the algebraic representation of a piecewise-linear function; graph it over the appropriate domain.
2. Write an algebraic representation for a given piecewise-linear function
3. Determine vertex, slope of each branch, intercepts, and end behavior of an absolute value graph.
4. Recognize and solve problems that can be modeled using absolute value, step, and other piecewise-linear functions.

   Examples: Postage rates, cellular telephone charges, tax rates.

F. Linear Equations, Inequalities and Systems of Linear Equations

Just as with linear functions, solving linear equations and inequalities in one variable should be reinforced as necessary before extending those skills to literal equations, equations, and inequalities involving absolute values, linear inequalities in two variables, and systems of equations and inequalities.

Successful students will:

F1 Solve linear and simple nonlinear equations involving several variables for one variable in terms of the others.

Example: Solve A = πr²h for h or for r.

F2 Solve equations and inequalities involving the absolute value of a linear expression in one variable.

F3 Graph the solution of linear inequalities in two variables.

1. Know what it means to be a solution of a linear inequality in two variables, represent solutions algebraically and graphically, and provide examples of ordered pairs that lie in the solution set.
2. Graph a linear inequality in two variables and explain why the graph is always a half-plane (open or closed).

F4 Solve systems of linear equations in two variables using algebraic procedures.

F5 Create, interpret, and apply mathematical models to solve problems arising from contextual situations that involve linear relationships.

1. Distinguish relevant from irrelevant information, identify missing information, and find what is needed or make appropriate estimates.
2. Apply problem solving heuristics to practical problems: Represent and analyze the situation using symbols, graphs, tables, or diagrams; assess special cases; consider analogous situations; evaluate progress; check the reasonableness of results; and devise independent ways of verifying results.
3. Recognize and solve problems that can be modeled using linear inequalities in two variables or a system of linear equations in two variables; interpret the solution(s) in terms of the context of the problem.

   Examples: Time/rate/distance problems; problems involving percentage increase or decrease; break-even problems, such as those comparing costs of two services; optimization problems that can be approached through linear programming.

4. Represent linear relationships using tables, graphs, verbal statements and symbolic forms; translate among these forms to extract information about the relationship.

F6 Determine, interpret, and compare linear models for data that exhibits a linear trend.

1. Identify and evaluate methods of determining the goodness of fit of a linear model.

   Examples: A linear model might pass through the most points, minimize the sum of the absolute deviations, or minimize the sum of the square of the deviations.

2. Use a computer or a graphing calculator to determine a linear regression equation (least-squares line) as a model for data that suggests a linear trend.
3. Determine and interpret correlation coefficients.
4. Use and interpret residual plots to assess the goodness of fit of a regression line.
5. Note the effect of outliers on the position and slope of the regression line.
6. Interpret the slope and y-intercept of the regression line in the context of the relationship being modeled.

G. Exponents and Simple Exponential Functions

The work done earlier with numerical expressions involving exponents is extended here to expressions involving variables. Variable expressions involving negative integral exponents are linked to students’ earlier work with reciprocal functions. Once solid understanding of the properties of exponents is established, students are introduced to exponential functions, that is, to functions in which the independent variable appears in the exponent. While students will have encountered applications involving exponential behavior as early as middle school, in this course students will be expected to gain facility over the symbolic and graphic representation of such relationships in simple cases. Algebra I exponential functions, along with reciprocal functions, provide students with non-linear examples through which they can come to a deeper understanding of linear behavior. More complex exponential applications, the solution of exponential equations and the relationship between exponential and logarithmic functions will be addressed in Algebra II.

Successful students will:

G1 Apply the properties of exponents to transform variable expressions involving integral and rational exponents.

1. Translate between rational exponents and notation involving integral powers and roots.

   Examples: a^p · a^q = a^p+q; ; 9^x = 3^2x; (8b⁶)^1/3 = 2b²; .

2. Factor out common factors with exponents.

   Examples: 6v⁷ + 12v⁵ - 8v³ = 2v³ (3v⁴ + 6v² - 4); 3x(x + 1)² - 2(x + 1)² = (x + 1)²(3x - 2).

   "Chunking" is a term often used to describe treating an expression, such as the
   x + 1 above as a single entity.

G2 Graph and analyze exponential functions and identify their key characteristics.

1. Identify functions having the general form f(x) = ab^x + c for b > 0, b ≠ 1 as exponential functions.
2. Recognize and represent the graphs of exponential functions; identify upper or lower limits (asymptotes).

G3 Recognize problems that can be modeled using exponential functions; interpret the solution(s) in terms of the context of the problem.

Exponential functions model situations where change is proportional to the initial quantity.

1. Use exponential functions to represent growth functions such as
   f(x) = an^x (a > 0 and n > 1) and decay functions such as
   f(x) = an^-x (a > 0 and n > 1).
2. Use the laws of exponents to determine exact solutions for problems involving exponential functions where possible; otherwise approximate the solutions graphically or numerically.

H. Quadratic Functions and Equations Over the Real Numbers

The final topic in Algebra I introduces yet another class of functions, those involving quadratic behavior. While students may have already encountered the classic application of quadratic functions, behavior of an object under the influence of gravity, Algebra I should present them with their first experience with general quadratic phenomena. This early work with quadratic functions rests on understanding and solving equations and should focus on those with real zeros/roots.

Successful students will:

H1 Identify quadratic functions expressed in multiple forms; identify the specific information each form clarifies.

1. Express a quadratic function as a polynomial, f(x) = ax² + bx + c, where a, b, and c are constants with a ≠ 0, and identify its graph as a parabola that opens up when a > 0 and down when a < 0; relate c to where the graph of the function crosses the y-axis.
2. Express a quadratic function having integral roots in factored form, f(x) = (x - r)(x - s), when r and s are integers; relate the factors to the solutions of the equation (x - r)(x - s) = 0 (x = r and x = s) and to the points ((r, 0) and (s, 0)) where the graph of the function crosses the x-axis.

H2 Graph quadratic functions and use the graph to help locate zeros.

A zero of a quadratic function f(x) = ax² + bx + c is a value of x for which f(x) = 0.

1. Sketch graphs of quadratic functions using both a graphing calculator and tables of values.
2. Estimate the real zeros of a quadratic function from its graph and identify quadratic functions that do not have real zeros by the behavior of their graphs.

H3 Recognize contexts in which quadratic models are appropriate; determine and interpret quadratic models that describe quadratic phenomena.

Examples: The relationship between length of the side of a square and its area; the relationship between time and distance traveled for a falling object.

H4 Solve quadratic equations with integral solutions; use quadratic equations to represent and solve problems involving quadratic behavior.

1. Solve quadratic equations that can be easily transformed into the form (x - a)(x - b) = 0 or (x + a)² = b, for integers a and b.
2. Estimate the roots of a quadratic equation from the graph of the corresponding function.

H5 Rewrite quadratic functions and interpret their graphical forms.

1. Write a quadratic function in polynomial or standard form,
   f(x) = ax² + bx + c, to identify the y-intercept of the function's parabolic graph or the x-coordinate of its vertex, .
2. Write a quadratic function in factored form, f(x) = a(x – r)(x – s), to identify its roots.
3. Write a quadratic function in vertex form, f(x) = a(x – h)² + k, to identify the vertex and axis of symmetry of the function's parabolic graph.
4. Determine domain and range, intercepts, axis of symmetry, and maximum or minimum.

H6 Graph quadratic equations and solve those with real solutions using a variety of methods.

1. Solve quadratic equations having real solutions by factoring, by completing the square, and by using the quadratic formula.
2. Use a calculator to approximate the roots of a quadratic equation and as an aid in graphing.
3. Select and explain a method of solution (e.g., exact vs. approximate) that is effective and appropriate to a given problem.
4. Recognize and solve practical problems that can be expressed using quadratic equations having real solutions; interpret their solutions in terms of the context of the situation.

   Examples: Determine the height of an object above the ground t seconds after it has been thrown upward from a platform d feet above the ground at an initial velocity of v₀ feet per second; find the area of a rectangle with perimeter 120 in terms of the length, L, of one side.

H7 Make regular fluent use of basic algebraic identities such as
(a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b²; and (a + b)(a - b) = a² - b².

1. Use the distributive law to derive each of these formulas.

   Examples: (a + b)(a - b) = (a + b)a - (a + b)b = (a² + ab) - (ab + b²) = a² + ab - ab - b² = a² - b²; applying this to specific numbers, 37 · 43 = (40 - 3)(40 + 3) = 1,600 - 9 = 1,591.

2. Use geometric constructions to illustrate these formulas.

   Example: Use a partitioned square or tiles to provide a geometric representation of (a + b)² = a² + 2ab + b².

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

Supporting Resources

Home

Jump to:

A. Applications of Numbers

B. Mathematical Vocabulary and Logic

C. Properties of Irrational and Real Numbers

D. Patterns of Growth through Iteration

E. Linear, Proportional and Piecewise-linear Functions

F. Linear Equations, Inequalities and Systems of Linear Equations

G. Exponents and Simple Exponential Functions

H. Quadratic Functions and Equations over the Real Numbers

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