Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Solve \(\sqrt{x+3}+\sqrt{x-1}=4\). State the domain before you move anything, and prove there is only one real solution.
Timer not started. Twenty minutes of real work before a solution.
Domain: \(x\ge 1\). Set \(a=\sqrt{x+3}\), \(b=\sqrt{x-1}\), so \(a+b=4\) and \(a^2-b^2=4\). Then \((a-b)(a+b)=4\), hence \(a-b=1\). Solving the linear pair gives \(a=5/2\), \(b=3/2\), so \(x=1+(3/2)^2=13/4\). The sum of two increasing square roots is strictly increasing on \([1,\infty)\), so the solution is unique. Check: \(\sqrt{25/4}+\sqrt{9/4}=5/2+3/2=4\).
Attack 2
The polynomial \(x^3-6x^2+11x-6\) has three integer roots. Find them. Then say, in one sentence, why the rational root theorem means you did not miss a rational root.
Timer not started. Twenty minutes of real work before a solution.
Possible rational roots are the factors of 6: \(\pm 1,2,3,6\). Evaluate: \(p(1)=0\), \(p(2)=0\), \(p(3)=0\). So \(p(x)=(x-1)(x-2)(x-3)\). Any rational root, written in lowest terms \(p/q\), must have \(p\) dividing the constant term and \(q\) dividing the leading coefficient. Here the leading coefficient is 1, so every rational root is an integer divisor of 6, and those were all tested.
Attack 3
Find the equation of the circle through \((0,0)\), \((4,0)\), and \((0,2)\).
Timer not started. Twenty minutes of real work before a solution.
Write \(x^2+y^2+Dx+Ey+F=0\). The origin forces \(F=0\). The point \((4,0)\) forces \(16+4D=0\), so \(D=-4\). The point \((0,2)\) forces \(4+2E=0\), so \(E=-2\). The circle is \(x^2+y^2-4x-2y=0\), center \((2,1)\), radius \(\sqrt{5}\).
Prerequisites
None. This is the start.
What this actually is
AP Precalculus is the last course in which a wrong algebraic move can still look like a small slip. After this, every later subject assumes you can rewrite an expression without changing its domain, read a graph as a set of pairs, and keep a parameter straight. The College Board course is a checklist of that fluency: polynomial and rational functions, exponential and logarithmic functions, trigonometric and polar functions, and functions involving parameters, vectors, and matrices at a light level. Passing the exam is not the target. The target is that none of those manipulations ever becomes the thing you are thinking about while you are trying to think about a proof or a derivative.
The hard part at this level is not a new theorem. It is domain, extraneous roots, and the difference between an identity and an equation that happens to hold at two points. A competition problem will hide a restriction inside a substitution. If you cannot say, out loud, what values were legal before you squared both sides, you are not done with this page.
You already sit in this course. Treat the class as a lower bound. The pages after this one will not reteach factoring, the unit circle, or logarithm rules. If one of those still costs you time, it is still this topic.
Core results
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Function. A function \(f:D\to C\) assigns to each \(x\in D\) exactly one value \(f(x)\in C\). The graph is the set of pairs. \(f\) is invertible precisely when it is one-to-one, and then \((f^{-1}\circ f)(x)=x\) on the domain.
If \(f(a)=f(b)\) implies \(a=b\), each value in the image has one preimage, so the inverse relation is a function. The two composition identities are the definition of that inverse. If some value is hit twice, the inverse relation fails to be single-valued.
Polynomials. If \(p\) is a polynomial and \(p(a)=0\), then \(x-a\) divides \(p(x)\). A nonconstant polynomial of degree \(n\) has at most \(n\) roots unless it is identically zero. For \(p(x)=a_n x^n+\cdots+a_0\) with integer coefficients, any rational root in lowest terms \(r/s\) has \(r\mid a_0\) and \(s\mid a_n\).
Divide by \(x-a\): \(p(x)=(x-a)q(x)+r\) with \(r\) constant, and \(p(a)=r\). The root bound is induction on degree after one linear factor is peeled off. The rational root theorem follows by clearing denominators and using that a prime dividing \(s\) cannot divide \(r\), hence cannot divide \(a_0\) after the equation is arranged; the symmetric argument handles \(r\) and \(a_n\).
Exponentials and logarithms. For \(b>0\), \(b\ne 1\), and positive \(x,y\): \(\log_b(xy)=\log_b x+\log_b y\), \(\log_b(x^k)=k\log_b x\), and \(\log_b x=\ln x/\ln b\). The exponential \(b^x\) is the inverse of \(\log_b\).
These are the exponent laws rewritten in inverse language. If \(x=b^u\) and \(y=b^v\), then \(xy=b^{u+v}\), which is the addition rule. Change of base is \(\ln x=\ln(b^{\log_b x})=\log_b x\cdot\ln b\).
Unit circle. On the circle \(x^2+y^2=1\), the point at angle \(\theta\) from the positive \(x\)-axis is \((\cos\theta,\sin\theta)\). Thus \(\cos^2\theta+\sin^2\theta=1\). Addition: \(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\) and \(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\).
The identity is the circle equation at that point. One clean derivation of addition uses Euler’s formula \(e^{i\theta}=\cos\theta+i\sin\theta\) and multiplies; a purely geometric derivation uses the distance formula between points at angles \(\alpha\) and \(-\beta\) on the unit circle and expands. Either is acceptable if you can reproduce it from a blank page.
Triangles and sequences. In a triangle, \(a/\sin A=b/\sin B=2R\) and \(c^2=a^2+b^2-2ab\cos C\). An arithmetic sequence has constant difference; a geometric sequence has constant ratio. The partial sum of a finite geometric series with ratio \(r\ne 1\) is \(a(1-r^{n})/(1-r)\).
Drop an altitude to prove the area formula \(\tfrac12 ab\sin C\), then equate two expressions for the area to get the law of sines; the law of cosines is the Pythagorean theorem applied to the two right triangles formed by an altitude. The geometric sum is the usual multiplication by \(1-r\).
The ladder
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Tier 0 / First contact
Watch or read once. Do not study from this.
Khan Academy, AP Precalculus Watch once, do not study from this. It is the official unit map, useful for seeing the shape of the course, and a bad place to spend your evenings.
Tier 1 / Main text
The thing you work through.
Stitz and Zeager, Precalculus, version 4−ε (15 January 2024) Free legal PDF from the authors. This is the book you work. OpenStax Precalculus 2e is the alternate if you want a second explanation of the same material: https://openstax.org/details/books/precalculus-2e
Tier 2 / Hard problems
The problem book.
Art of Problem Solving, PrecalculusPAID The competition-flavored textbook. Paid. Use it after Stitz and Zeager, not instead of it. Alcumus at a hard rating is the daily drill: https://artofproblemsolving.com/alcumus
Tier 3 / The ceiling
The hardest questions that exist at this level.
Po-Shen Loh, live contest index, and the AMC 12 / AIME archives Real MAA problems, sorted by concept, with difficulty ratings. The AoPS indexes, opened and checked in a browser, are https://artofproblemsolving.com/wiki/index.php/AMC_12_Problems_and_Solutions and https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions . Learn Anything has no precalculus page.
Hardest questions
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: five consecutive AMC 12 problems numbered 21 through 25, chosen from five different years, all on functions, algebra, or trigonometry, solved cold. Full written solutions. No calculator unless the contest allowed one. Fail if any solution uses a step you cannot justify, including a squared equation whose extra roots you did not throw out.
Diagrams
The unit circle is a definition, not a mnemonic. Cosine and sine are coordinates.
Escalating sets
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Set 1. Cleanups
State the domain of \(\log_2(x^2-1)\) and of \(\sqrt{4-x}+\sqrt{x+1}\).
Solve \(2^{x}+2^{x+1}+2^{x+2}=7\cdot 2^{x}\) and explain the cancellation.
Expand \(\sin(2\theta)\) and \(\cos(2\theta)\) from the addition formulas. Do not quote them from memory first.
Find the inverse of \(f(x)=(2x-1)/(x+3)\) and its domain.
Sum \(3+6+12+\cdots+3\cdot 2^{7}\).
Set 2. Restrictions
Solve \(\sqrt{x+5}=x-1\) and list every extraneous root you created.
Solve \(\log_2(x-1)+\log_2(x+1)=3\).
For which \(k\) does \(x^2-2x+k=0\) have two distinct real roots? Draw the condition.
A rational function has vertical asymptotes \(x=1\) and \(x=-2\) and a horizontal asymptote \(y=2\). Write one such function and one that fails the horizontal condition.
Prove that \(\log_b a\cdot\log_a b=1\) for legal bases and arguments.
Set 3. Graphs as sets
Sketch \(y=2\sin(x-\pi/3)+1\) by listing the transformations in the order you apply them.
How many solutions does \(\sin x=x/10\) have? Argue from graphs, no calculator.
Find the range of \(f(x)=(x^2+1)/(x^2+2)\).
Parameterize the line through \((1,2)\) and \((4,-1)\). Where does it meet the circle \(x^2+y^2=1\)?
Show that \(x+1/x\ge 2\) for \(x>0\), with equality only at \(x=1\).
Set 4. Contest shape
Find all real \(x\) with \(x^2+x+1\) dividing \(x^3-1\), and factor the quotient.
Solve \(\sin\theta+\cos\theta=1\) exactly, on \([0,2\pi)\).
The polynomial \(x^4+4\) factors over the reals. Find the factorization.
A geometric sequence of positive terms has second term 6 and fourth term 24. Find the first term. How many answers?
From Po-Shen Loh, do five problems rated in the top band under polynomials or logs. Write them up.
Set 5. Ceiling
Five AMC 12 problems numbered 15–25, mixed years, written in full.
Three AIME problems numbered 8–15 on algebra or trig.
Ten Sharygin problems that are solved by coordinates, not by a pure Euclidean theorem you have not proved.
Rebuild the unit-circle addition formula without looking.
One Alcumus session at the hardest available setting, topic of your weakest function family. Keep the written work, not the score.
Return bank
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
\(\sqrt{x+3}+\sqrt{x-1}=4\), with the uniqueness argument.
Rational roots of \(x^3-6x^2+11x-6\), and the theorem that closed the search.
Circle through \((0,0)\), \((4,0)\), \((0,2)\).
Domain of \(\log_2(x^2-1)\).
\(\sin\theta+\cos\theta=1\) on one full turn.
Factor \(x^4+4\) over the reals.
Range of \((x^2+1)/(x^2+2)\).
2023 AMC 12A #21, written, not watched.
One AIME #12 on logs or trig, from the archive.
Law of cosines derived from an altitude, not quoted.
Oral pass
From a blank page, define a function, state the rational root theorem, derive \(\sin^2+\cos^2=1\) from the circle, and solve one logarithm equation while naming the domain out loud.