Precalculus 2e PDF Download – OpenStax (BS Math/BSCS)

Students taking Precalculus can download the complete textbook “Precalculus 2e” by OpenStax, free as a PDF or web-reader edition from OpenStax’s own site. It is a 12-chapter bridge course between algebra and calculus — functions, polynomial and rational functions, exponential and logarithmic functions, trigonometry, systems of equations, analytic geometry, and sequences — that closes with its own short introduction to limits and derivatives.

For BS Mathematics and BSCS students, this is usually the course that either confirms algebra fluency before Calculus I or fills the specific gaps — trigonometric identities, conic sections, matrix systems — that a weaker algebra background left behind. Every section ends with a full practice-problem set and every chapter closes with its own Key Terms, Key Equations, Key Concepts, and Chapter Review Exercises, which makes the book usable for board-exam-style self-testing as well as first reading.

Book Overview

Chapter List

Chapter 1: Functions

Difficulty: Medium · Key topics: function notation, domain & range, average rate of change, composition, transformations, absolute value functions, inverse functions

This opening chapter builds the vocabulary the rest of the book depends on. It uses the dot-com-era S&P 500 story — a real relationship between year and stock average — to show why data gets described as functions in the first place, then formalizes what makes a relation a function, how to read function notation, and how to find a function’s domain and range from its equation or graph. It moves on to measuring a function’s average rate of change and reading increasing/decreasing behavior off a graph, then covers composing two functions, transforming a parent function (shifts, stretches, reflections), absolute value functions as a special case, and finally how to find and verify a function’s inverse.

Key Points:

• A relation is a function only if every input maps to exactly one output — check with the vertical line test on a graph
• Function notation f(x) reads as “f of x”, the output when rule f is applied to input x, not f times x
• Average rate of change over [a,b] is (f(b) − f(a)) / (b − a) — the slope of the secant line
• Composition (f ∘ g)(x) = f(g(x)) applies g first, then f — order matters
• Transformations: f(x) + k shifts vertically, f(x + h) shifts horizontally (opposite the sign), af(x) stretches/reflects vertically
• A function has an inverse only where it is one-to-one; verify with f(f⁻¹(x)) = x

Memory Tip: Read f(x+h) transformations backwards from what you’d expect — a “+h” inside the parentheses shifts the graph LEFT, not right. Say it out loud as “inside is opposite” to remember it.

Common Mistake: Confusing f(x) + k (moves the whole graph up/down) with f(x + k) (moves it left/right), because both look like simple addition — always check whether the constant is inside or outside the function’s parentheses before deciding the direction.

Important Questions:

• What test determines whether a graph represents a function? The vertical line test: if any vertical line crosses the graph more than once, it is not a function.
• How do you find the average rate of change of a function between two points? Compute (f(b) − f(a)) / (b − a), the slope of the line connecting those two points on the graph.

Chapter 2: Linear Functions

Difficulty: Easy · Key topics: slope, slope-intercept form, point-slope form, parallel & perpendicular lines, linear modeling, linear regression

Using a bamboo species that grows nearly 1.5 inches every hour as its opening example, this chapter formalizes what makes a function linear: a constant rate of change, expressed as slope. It covers writing and interpreting linear equations in slope-intercept and point-slope form, graphing lines from an equation and vice versa, and identifying when two lines are parallel (equal slopes) or perpendicular (negative reciprocal slopes). The chapter then turns practical, building linear models directly from a word problem’s constant rate, and closes by fitting a linear model to real, scattered data using regression rather than a perfect straight-line relationship.

Key Points:

• Slope m = (y2 − y1)/(x2 − x1) measures constant rate of change; a linear function has the same slope everywhere
• Slope-intercept form: y = mx + b, where b is the y-intercept
• Point-slope form: y − y1 = m(x − x1), useful when you know one point and the slope but not the intercept
• Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals of each other
• A linear model from a word problem: slope = the constant rate given, intercept = the starting value
• Linear regression finds the best-fit line through scattered real data, not an exact fit through every point

Memory Tip: To keep point-slope form straight, remember it starts from a known POINT: y − y₁ = m(x − x₁) — both subtractions use the coordinates of that one point you were given.

Common Mistake: Mixing up which slope condition applies to perpendicular lines — students often use “equal slopes” for perpendicular and “negative reciprocal” for parallel, the reverse of the actual rule.

Important Questions:

• How do you know two lines are perpendicular from their equations? Their slopes are negative reciprocals of each other (e.g., 2 and −1/2); if the product of the two slopes is −1, the lines are perpendicular.
• What does the y-intercept represent in a real-world linear model? The starting value of the quantity being modeled, at x = 0, before any time or input has passed.

Chapter 3: Polynomial and Rational Functions

Difficulty: Medium · Key topics: complex numbers, quadratic functions, end behavior, polynomial division, zeros, rational functions, asymptotes, variation

Framed by a scuba-diving scenario about gas laws and pressure at depth, this long chapter builds the full polynomial toolkit. It opens with complex numbers (needed once quadratics stop having real roots), then covers quadratic functions in vertex and standard form, general power and polynomial functions and how degree and leading coefficient control end behavior, and reading multiplicities and turning points off a polynomial’s graph. From there it covers dividing polynomials (long and synthetic division), finding real and complex zeros using the Rational Zero Theorem and Descartes’ Rule of Signs, graphing rational functions around their asymptotes and holes, inverse and radical functions, and closes with direct, inverse, and joint variation models.

Key Points:

• i = √−1, and i² = −1; complex numbers a + bi let every quadratic have a solution, real or not
• A quadratic in vertex form f(x) = a(x − h)² + k has vertex (h, k); the sign of a sets whether it opens up or down
• End behavior of a polynomial is set by its degree (even/odd) and leading coefficient’s sign — the highest-degree term dominates for large |x|
• A zero’s multiplicity determines whether the graph crosses the x-axis there (odd) or bounces off it (even)
• Rational functions have vertical asymptotes where the denominator is zero (numerator isn’t) and holes where both are zero at the same point
• Direct variation: y = kx; inverse variation: y = k/x — k is the constant of variation, found from one known pair

Memory Tip: For end behavior, remember “leading term wins” — plug in a very large x mentally, and only the highest-degree term matters for the graph’s far-left and far-right behavior.

Common Mistake: Treating a hole (removable discontinuity) and a vertical asymptote as the same thing — a hole happens when a factor cancels from both numerator and denominator; an asymptote happens when the denominator’s factor doesn’t cancel.

Important Questions:

• How do you find the vertical asymptotes of a rational function? Set the denominator equal to zero and solve; any x-value that makes the denominator (but not the numerator) zero at the same point is a vertical asymptote.
• What determines whether a polynomial’s graph crosses or bounces off the x-axis at a zero? The zero’s multiplicity: odd multiplicity crosses the x-axis, even multiplicity touches and bounces back.

Chapter 4: Exponential and Logarithmic Functions

Difficulty: Medium · Key topics: exponential growth & decay, the number e, logarithms, log properties, exponential/log equations, exponential modeling

Opening with the fact that bacterial cells vastly outnumber human cells in the body, this chapter uses biological growth to motivate exponential functions: their graphs, growth versus decay, and the natural base e. It then introduces logarithms as the inverse of exponentiation, graphing logarithmic functions and their properties (product, quotient, power rules, and change-of-base). The second half is applied: solving exponential and logarithmic equations algebraically, building exponential growth/decay and logistic models from a real scenario, and fitting an exponential curve to scattered real-world data the way Chapter 2 fit a line.

Key Points:

• An exponential function f(x) = ab^x, with b > 0, b ≠ 1; b > 1 models growth, 0 < b < 1 models decay
• The number e ≈ 2.71828 is the natural base, used in continuous growth/decay: A = Pe^(rt)
• log_b(x) = y means b^y = x — a logarithm is the exponent needed on base b to get x
• Log properties: log_b(mn) = log_b(m) + log_b(n); log_b(m/n) = log_b(m) − log_b(n); log_b(m^p) = p·log_b(m)
• Change-of-base formula: log_b(x) = ln(x)/ln(b), needed for any base a calculator doesn’t have a button for
• Exponential graphs have a horizontal asymptote; logarithmic graphs have a vertical asymptote, where their input would hit zero

Memory Tip: Logarithms and exponentials undo each other exactly like squaring and square-rooting — a variable in an exponent means take a log of both sides; a variable inside a log means exponentiate both sides.

Common Mistake: Applying log_b(m + n) = log_b(m) + log_b(n) — there is no such rule. The product rule only works for log_b(m · n), not for a sum inside a single logarithm.

Important Questions:

• How do you solve an exponential equation like 3^x = 20? Take the natural log of both sides: x = ln(20)/ln(3), then evaluate with a calculator.
• What is the difference between exponential growth and decay in f(x) = ab^x? Growth occurs when base b > 1 (the function increases as x increases); decay occurs when 0 < b < 1 (the function decreases as x increases).

Chapter 5: Trigonometric Functions

Difficulty: Medium · Key topics: radians vs degrees, coterminal & reference angles, unit circle, the six trig functions, right-triangle trigonometry

Using tidal patterns and how Torres Strait Islander fishers historically timed their work around them, this chapter introduces trigonometry as the mathematics of things that repeat in regular cycles. It covers measuring angles in degrees and radians, converting between them, and finding coterminal and reference angles. The unit circle is then built from scratch to define sine and cosine for any angle, not just acute ones, followed by the remaining four trig functions defined from sine and cosine. The chapter closes by connecting all of this back to right-triangle trigonometry (SOHCAHTOA) for angles between 0° and 90°.

Key Points:

• One full revolution = 360° = 2π radians; convert with degrees × (π/180) = radians, and radians × (180/π) = degrees
• A reference angle is the acute angle between the terminal side and the x-axis — used to find exact trig values in any quadrant
• On the unit circle, cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the point at angle θ
• tan(θ) = sin(θ)/cos(θ); reciprocal functions are csc = 1/sin, sec = 1/cos, cot = 1/tan
• SOHCAHTOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent — right triangles only
• Sign of a trig function depends on the quadrant — tracked by “All Students Take Calculus” (ASTC)

Memory Tip: “All Students Take Calculus” — going counter-clockwise from quadrant I, this names which function is positive in each quadrant: All (I), Sine (II), Tangent (III), Cosine (IV).

Common Mistake: Leaving a calculator in the wrong angle mode — using degree mode when a problem is in radians (or vice versa) gives a numerically valid but completely wrong answer, the single most common trig-chapter error.

Important Questions:

• How do you convert 45° to radians? Multiply by π/180: 45 × (π/180) = π/4 radians.
• What are the coordinates of the point on the unit circle at angle θ? (cos θ, sin θ) — cosine gives the x-coordinate, sine gives the y-coordinate.

Chapter 6: Periodic Functions

Difficulty: Medium · Key topics: amplitude, period, phase shift, vertical shift, graphing tan/cot/sec/csc, inverse trig functions

Building on the sun’s daily and yearly cycle — the chapter opens with how ancient Egyptian, Hindu, and Native American traditions all tracked time by the sun’s regular motion — this chapter is entirely about graphing periodic functions. It covers how amplitude, period, phase shift, and vertical shift transform the basic sine and cosine graphs, then extends the same graphing skills to tangent, cotangent, secant, and cosecant, each with its own asymptote pattern. It closes with inverse trigonometric functions: arcsin, arccos, and arctan, and why each needs a restricted domain on the original function before an inverse can exist at all.

Key Points:

• For y = A sin(B(x − C)) + D: amplitude = |A|, period = 2π/|B|, phase shift = C, vertical shift = D
• Tangent and cotangent have period π (not 2π) — they repeat twice as often as sine and cosine
• Secant and cosecant have vertical asymptotes wherever their reciprocal function (cosine or sine) equals zero
• arcsin and arccos have restricted range: arcsin outputs [−π/2, π/2], arccos outputs [0, π]
• arctan has range (−π/2, π/2) and is defined for all real numbers, unlike arcsin/arccos which need input in [−1, 1]
• sin⁻¹(x) is notation for arcsin(x), NOT 1/sin(x) — the two look alike but mean very different things

Memory Tip: To sketch any sine/cosine graph fast, find amplitude, period, phase shift, and vertical shift first, plot the midline and one full cycle, then repeat the pattern — don’t plot point by point.

Common Mistake: Reading sin⁻¹(x) as “1 over sin(x)” — it is the inverse function (arcsin), not the reciprocal. The reciprocal of sine is cosecant, csc(x) = 1/sin(x).

Important Questions:

• How do you find the period of y = 3 sin(2x)? Period = 2π/|B| = 2π/2 = π; this graph completes one full cycle every π units.
• Why does arcsin(x) need a restricted domain and range to exist as a function? Sine is not one-to-one over its full domain, so its range is restricted to [−π/2, π/2] to make the inverse pass the vertical line test.

Chapter 7: Trigonometric Identities and Equations

Difficulty: Hard · Key topics: Pythagorean identities, sum/difference formulas, double/half-angle formulas, solving trig equations, modeling periodic data

Framed by how sinusoidal graphs model recorded sound and music, this is the most algebra-heavy chapter in the book. It starts with the fundamental (Pythagorean, even/odd, reciprocal) identities and the technique of verifying one side of an equation equals the other. It then builds the sum and difference identities for sine, cosine, and tangent, derives the double-angle, half-angle, and power-reduction formulas from them, and covers the less-common sum-to-product and product-to-sum conversions. The chapter closes by solving trigonometric equations for all solutions in a given interval, and modeling real periodic phenomena with a fitted sinusoidal function.

Key Points:

• Pythagorean identity: sin²θ + cos²θ = 1; dividing by cos²θ or sin²θ gives the related tan/sec and cot/csc identities
• Sum identity: sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB (note the sign flip)
• Double-angle: sin(2θ) = 2 sinθ cosθ; cos(2θ) has three equivalent forms depending on which identity you substitute
• Verifying an identity means transforming one side until it matches the other — never treat it as an equation to solve
• Solving trig equations for “all solutions” requires adding +2πn (or +πn for tangent) to capture every coterminal answer
• A sinusoidal model fits real periodic data using the same amplitude/period/phase/vertical-shift parameters from Chapter 6

Memory Tip: When stuck simplifying an identity, convert everything to sine and cosine first — the algebra usually simplifies once there’s only two functions to juggle.

Common Mistake: Treating “verify this identity” like an equation to solve — students cross-multiply or move terms across the equals sign, which is invalid until the identity is actually proven true.

Important Questions:

• What is the Pythagorean identity and where does it come from? sin²θ + cos²θ = 1, which comes directly from the unit circle definition where (cosθ, sinθ) is a point at distance 1 from the origin.
• Why do trigonometric equations usually have infinitely many solutions? Sine, cosine, and tangent are periodic, so any solution θ has infinitely many coterminal angles (θ + 2πn or θ + πn) that also satisfy the equation.

Chapter 8: Further Applications of Trigonometry

Difficulty: Hard · Key topics: Law of Sines/Cosines, polar coordinates, polar form of complex numbers, parametric equations, vectors

Opening with how scientists measure the height of General Sherman, the world’s largest tree by volume, using angle-of-elevation trigonometry instead of a tape measure, this chapter extends trig beyond right triangles and rectangular coordinates. It covers the Law of Sines and Law of Cosines for solving any triangle, then pivots to polar coordinates, their graphs, and how complex numbers can be written and multiplied more easily in polar form. It closes with parametric equations (x and y both functions of a third variable t) and their graphs, and vectors — magnitude, direction, addition, and the dot product.

Key Points:

• Law of Sines: a/sinA = b/sinB = c/sinC — use when you know an angle and its opposite side, plus one more piece
• Law of Cosines: c² = a² + b² − 2ab cosC — use for SAS or SSS triangles where Law of Sines can’t start
• A polar point (r, θ) converts to rectangular with x = r cosθ, y = r sinθ
• Polar form of a complex number: z = r(cosθ + i sinθ) — makes multiplying/finding powers of complex numbers far easier than a + bi form
• Parametric equations x(t), y(t) trace a path over “time” t — the same curve can look very different depending on how t is chosen
• A vector’s magnitude is ‖v‖ = √(x²+y²); the dot product u·v = u₁v₁ + u₂v₂ tests whether two vectors are perpendicular (dot product = 0)

Memory Tip: Ambiguous-case triangles (Law of Sines, SSA) can have 0, 1, or 2 valid solutions — always sketch the triangle and check it actually closes before accepting an answer.

Common Mistake: Forgetting that a polar point (r, θ) is not unique — the same location can be written with a negative r or with θ + 2πn, which trips students up when converting or graphing.

Important Questions:

• When should you use the Law of Cosines instead of the Law of Sines? When you know all three sides (SSS) or two sides and the included angle (SAS) — no angle-opposite-side pair is available to start the Law of Sines.
• How do you convert the polar point (4, π/3) to rectangular coordinates? x = 4cos(π/3) = 2, y = 4sin(π/3) = 2√3, so the rectangular point is (2, 2√3).

Chapter 9: Systems of Equations and Inequalities

Difficulty: Medium · Key topics: substitution & elimination, nonlinear systems, partial fractions, matrix operations, Gaussian elimination, Cramer’s Rule

This chapter’s own introduction promises the tools to solve systems of equations using matrices — structured algebra at scale, not code-breaking. It opens with solving two-variable linear systems by substitution and elimination, extends to three-variable systems, and covers nonlinear systems where a line and a curve (or two curves) intersect. Partial fraction decomposition follows as a technique that reverses combining fractions. The rest of the chapter is matrix-based: matrix operations, Gaussian elimination and row-echelon form, solving a system using a matrix’s inverse, and Cramer’s Rule for solving small systems using determinants.

Key Points:

• A linear system’s solution is where all the equations are simultaneously true — graphically, where the lines/curves intersect
• Elimination adds/subtracts multiples of equations to cancel a variable; substitution solves one equation for a variable and plugs it into another
• Matrix multiplication is NOT commutative: AB ≠ BA in general, and inner dimensions must match for the product to be defined
• Gaussian elimination reduces a matrix to row-echelon form using row operations, then back-substitutes to solve the system
• A matrix has an inverse only if its determinant is nonzero; A⁻¹A = I, the identity matrix
• Cramer’s Rule solves for each variable as a ratio of two determinants — fast for 2×2/3×3 systems, impractical for larger ones

Memory Tip: For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc — memorize this first, since Cramer’s Rule, inverses, and singularity all depend directly on it.

Common Mistake: Trying to “divide” by a matrix — matrices don’t have division; to solve AX = B you multiply both sides by A⁻¹ on the correct side, and matrix multiplication order matters.

Important Questions:

• How do you know a system of linear equations has no solution? Elimination or substitution produces a false statement (like 0 = 5); graphically, the lines are parallel and never intersect.
• What condition must be true for a square matrix to have an inverse? Its determinant must be nonzero; a matrix with determinant zero is singular and has no inverse.

Chapter 10: Analytic Geometry

Difficulty: Hard · Key topics: conic sections, ellipse, hyperbola, parabola, foci & directrix, rotation of axes, polar equations of conics

Using the geometric image of a cone sliced by a plane at different angles, this chapter derives every conic section from one shared idea before treating each separately in full algebraic detail. It covers the ellipse (two foci, sum of distances constant), the hyperbola (two foci, difference of distances constant), and the parabola (one focus and a directrix, distances equal) — their standard-form equations, how to find center, vertices, and foci directly from the equation, and how to write an equation from given geometric features. It closes with rotating coordinate axes to eliminate an xy-term in a general conic equation, and expressing conics in polar coordinates.

Key Points:

• Ellipse standard form: (x−h)²/a² + (y−k)²/b² = 1, a > b; foci lie on the major axis at distance c, where c² = a² − b²
• Hyperbola standard form: (x−h)²/a² − (y−k)²/b² = 1; foci satisfy c² = a² + b²
• Parabola: distance from any point on the curve to the focus equals its distance to the directrix — this rule generates the standard equation
• Rotation of axes removes the xy-term from a general second-degree equation, revealing which conic it actually represents
• The discriminant B² − 4AC classifies a general conic before you solve: negative = ellipse, zero = parabola, positive = hyperbola
• Polar equations of conics use eccentricity e: e < 1 is an ellipse, e = 1 is a parabola, e > 1 is a hyperbola

Memory Tip: Ellipse uses PLUS between the two squared terms (sum of distances); hyperbola uses MINUS (difference of distances) — the sign matches the geometric definition directly.

Common Mistake: Mixing up a² and b² when identifying an ellipse’s major axis — a² is always under the variable matching the LONGER axis, so a larger y-term denominator means a vertical major axis.

Important Questions:

• How do you find the foci of the ellipse x²/25 + y²/9 = 1? a² = 25, b² = 9, so c² = a² − b² = 16, c = 4; the foci are at (±4, 0) since the major axis is horizontal.
• What single geometric property defines every point on a parabola? Every point is equidistant from the focus and the directrix — that equal-distance condition is the definition the standard equation comes from.

Chapter 11: Sequences, Probability and Counting Theory

Difficulty: Medium · Key topics: sequences, arithmetic & geometric sequences, series, permutations & combinations, Binomial Theorem, probability

Opening with a lottery winner’s real choice between a lump sum and a 30-year annuity, this chapter uses that framing to motivate sequences and series as tools for modeling repeated, structured payments and patterns. It covers sequences defined explicitly or recursively, then arithmetic sequences (constant common difference) and geometric sequences (constant common ratio) and their sum formulas, including the special case of an infinite geometric series. Counting principles follow: the multiplication principle, permutations, and combinations, feeding directly into the Binomial Theorem for expanding (x+y)^n without multiplying it out term by term. The chapter closes with basic probability, built on the same counting tools.

Key Points:

• Arithmetic sequence: aₙ = a₁ + (n−1)d, where d is the constant common difference between consecutive terms
• Geometric sequence: aₙ = a₁·r^(n−1); a geometric series only converges (has a finite sum) if |r| < 1
• Sum of a finite arithmetic series: Sₙ = n(a₁+aₙ)/2; finite geometric series: Sₙ = a₁(1−rₙ)/(1−r)
• Permutations (order matters): nPr = n!/(n−r)!; combinations (order doesn’t matter): nCr = n!/(r!(n−r)!)
• The Binomial Theorem expands (x+y)ⁿ using combinations as coefficients: each term is C(n,k)·x^(n−k)·y^k
• Basic probability of an event = (favorable outcomes) / (total possible outcomes), and counting principles find both numbers

Memory Tip: To tell permutations from combinations fast, ask “does order matter?” — a race’s 1st/2nd/3rd place is a permutation; a 3-person committee is a combination.

Common Mistake: Using nPr when the problem actually describes a combination (or the reverse) — always identify first whether rearranging the same items counts as a different outcome.

Important Questions:

• What is the difference between an arithmetic and a geometric sequence? Arithmetic has a constant DIFFERENCE between consecutive terms (add the same number each time); geometric has a constant RATIO (multiply by the same number each time).
• When does an infinite geometric series have a finite sum? Only when the common ratio r satisfies |r| < 1; then the sum is a₁/(1−r). If |r| ≥ 1, the series diverges.

Chapter 12: Introduction to Calculus

Difficulty: Hard · Key topics: limits (numerical, graphical, algebraic), one-sided limits, continuity, the difference quotient, the derivative

Closing the book with a genuine bridge to Calculus I, this chapter opens with distance runner Sifan Hassan’s changing pace during a race to ask a concrete question: how fast is she going at one exact instant, not on average? It builds limits from the ground up — estimating them numerically from a table, reading them off a graph, and computing them algebraically using limit laws and techniques for indeterminate forms. Continuity is defined precisely in terms of limits, and the chapter ends by using the difference quotient and the limit definition to introduce the derivative as the exact, instantaneous version of the average rate of change from Chapter 1.

Key Points:

• A limit lim(x→a) f(x) = L describes the value f(x) approaches as x nears a — it does not require f(a) itself to exist or equal L
• One-sided limits (left and right) must both exist and be equal for the two-sided limit to exist
• A function is continuous at x = a only if f(a) is defined, lim(x→a) f(x) exists, and they’re equal
• The difference quotient (f(x+h) − f(x))/h is the average rate of change over a tiny interval of width h
• The derivative f'(x) = lim(h→0) (f(x+h) − f(x))/h is the limit of the difference quotient as h shrinks to zero
• The indeterminate form 0/0 in a limit often simplifies by factoring, rationalizing, or canceling before evaluating

Memory Tip: A limit asks “where is the function headed?”, not “where does it actually land?” — a function can have a hole exactly at x = a and still have a perfectly well-defined limit there.

Common Mistake: Assuming a limit doesn’t exist just because direct substitution gives 0/0 — that’s a signal to simplify algebraically (factor and cancel, or rationalize), not proof the limit fails to exist.

Important Questions:

• What three conditions must be true for a function to be continuous at x = a? f(a) must be defined, the limit as x approaches a must exist, and that limit must equal f(a).
• How is the derivative defined using the difference quotient? f'(x) = lim(h→0) [f(x+h) − f(x)] / h — the limit of the average rate of change over a shrinking interval h.

Download Precalculus 2e PDF (Free)

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How to Study This Book

Chapters 1 through 3 (Functions, Linear Functions, Polynomial and Rational Functions) are the algebra-review core of the book. If your function notation, factoring, or graphing is shaky, budget the most time here — every later chapter assumes fluency with it, and rushing this part is the single biggest reason students struggle from Chapter 5 onward.

Chapter 4 (Exponential and Logarithmic Functions) can be read somewhat independently of Chapters 5 through 8. If your course syllabus covers exponentials and logarithms before trigonometry, that order works fine with this book — the chapters don’t depend on each other.

Chapters 5 through 8 (Trigonometric Functions, Periodic Functions, Trigonometric Identities and Equations, Further Applications of Trigonometry) form one continuous unit. Do not skip Chapter 5 to get to Chapter 7 — the identities and equations in Chapter 7 depend entirely on the unit-circle definitions built in Chapter 5, and Chapter 8’s Law of Sines/Cosines and polar work depend on both.

Chapter 9 (Systems of Equations and Inequalities) and Chapter 10 (Analytic Geometry) are largely independent of the trigonometry block and of each other. Read them in either order, or return to them later, depending on how your own course sequences the material.

Chapter 11 (Sequences, Probability and Counting Theory) is self-contained and is often taught as a standalone unit near the end of a Precalculus course. BSCS-focused syllabi sometimes skip it here in favor of a dedicated Discrete Mathematics course — if yours does, this chapter is a solid quick review rather than new material.

Chapter 12 (Introduction to Calculus) is the book’s real payoff chapter. Read it only after finishing at least Chapters 1 through 8 — its limits and derivative examples lean directly on function notation, graph-reading, and (for a few examples) trigonometric limits.

What this book does not cover: it is a Precalculus text, not a Calculus text. Beyond Chapter 12’s introduction to limits, continuity, and the derivative definition, there are no differentiation rules, no integration, and no multivariable content. Students who need a full first-semester calculus course should move on to Calculus Volume 1 after finishing this book.

Used In These Programs

This book is used for the Precalculus course in: BS Mathematics · BS Computer Science. Browse all Calculus books or all Mathematics books.

Who Should Read This

Precalculus 2e is written for a student who has already finished high-school-level algebra and needs one more course to be fully ready for Calculus I — it is not a from-scratch introduction to algebra, and it moves quickly through material a strong algebra student already half-knows. It suits BS Mathematics and BSCS students taking a required Precalculus or Pre-Engineering Math course, and it is equally useful as a targeted refresher for anyone about to start Calculus Volume 1 who wants to shore up trigonometric identities, conic sections, or matrix systems first rather than relearning them mid-calculus-course. It is not the right choice for someone who has never seen function notation or basic algebra at all — that student needs an algebra text before this one.

Applicable Universities

This book is useful for students at Pakistani universities offering BS Mathematics and BS Computer Science programs, including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

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• Calculus Volume 1 – Strang & Herman (OpenStax)
• Calculus Volume 2 – Strang & Herman (OpenStax)
• Calculus Volume 3 – Strang & Herman (OpenStax)
• Discrete Mathematics: An Open Introduction – Oscar Levin

Precalculus 2e is this project’s direct bridge into its three-volume OpenStax Calculus series — the natural next step once you finish it is Calculus Volume 1. Browse more Mathematics books for the rest of your semester.

Precalculus 2e, by Jay Abramson and contributing authors. OpenStax, Rice University. Free under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence. Access for free at https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions