Core Concepts & Key Formulas
Review each topic before attempting the problems. Items marked Memorize must be recalled instantly in exams.
Chapter 01
Functions & Composition
- Domain: all valid inputs; Range: all outputs
- $(f \circ g)(x) = f(g(x))$ — apply $g$ first, then $f$ Memorize
- Vertical Line Test: each $x$ gives exactly one $y$
$f(f(a))$: compute $f(a)$ first, then substitute the result back into $f$.
Chapter 02
Inverse Functions
- $f^{-1}$ exists iff $f$ is one-to-one (passes Horizontal Line Test)
- To find $f^{-1}$: swap $x$ and $y$, then solve for $y$ Memorize
- $(f \circ f^{-1})(x) = x$
Chapter 03
Polynomials & Rational Expressions
- Rational Root Theorem: possible roots $= \pm\frac{p}{q}$ where $p\mid a_0,\; q\mid a_n$ Memorize
- Factor theorem: $(x-c)$ is a factor iff $f(c)=0$
- Simplify rational expressions: cancel common factors (state restrictions)
Chapter 04
Exponential & Logarithmic Functions
- $\log_b(xy)=\log_b x+\log_b y$; $\log_b\frac{x}{y}=\log_b x-\log_b y$ Memorize
- $\log_b(x^n)=n\log_b x$
- $b^x=b^y \Rightarrow x=y$; same-base equations
- Always check for extraneous solutions in log equations ($x>0$)
Change of base: $\log_b x = \dfrac{\ln x}{\ln b}$
Chapter 05
Sequences & Series
- Arithmetic: $a_n = a_1+(n-1)d$; $S_n=\frac{n}{2}(a_1+a_n)$ Memorize
- Geometric: $a_n=a_1 r^{n-1}$; $S_\infty=\frac{a_1}{1-r}$ for $|r|<1$ Memorize
Chapter 06
Trigonometry & Identities
- $\sin^2\theta+\cos^2\theta=1$; $1+\tan^2\theta=\sec^2\theta$ Memorize
- $\cos(2\theta)=1-2\sin^2\theta=2\cos^2\theta-1$ Memorize
- Reference angles: $\frac{\pi}{6}(30°),\frac{\pi}{4}(45°),\frac{\pi}{3}(60°)$
$\sin\frac{\pi}{6}=\frac{1}{2},\;\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2},\;\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}$
Chapter 07
Law of Sines & Cosines
- Law of Cosines: $c^2=a^2+b^2-2ab\cos C$ Memorize
- Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Chapter 08
Conic Sections
- Parabola: $y=a(x-h)^2+k$; vertex $(h,k)$ Memorize
- Ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$; foci $c^2=a^2-b^2$ ($a>b$) Memorize
- Hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$; asymptotes $y=\pm\frac{b}{a}x$ Memorize
Chapter 09
Matrices, Vectors & Complex Numbers
- $\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$; $\det(A^n)=(\det A)^n$ Memorize
- Dot product: $\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos\theta$
- Complex division: multiply numerator & denominator by conjugate
Chapter 10
Limits & Binomial Theorem
- $\lim_{x\to 0}\frac{\sin kx}{x}=k$ Memorize
- Binomial: $(a+b)^n=\sum_{k=0}^n\binom{n}{k}a^{n-k}b^k$ Memorize
- General term: $T_{k+1}=\binom{n}{k}a^{n-k}b^k$
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