Function

“A vending machine is a function: one button, one snack, every time.”

A rule assigning each input exactly one output.

Memory trick: One input → exactly one output.

Domain and range

“You can't feed a negative into a square root over the reals — that limits the domain.”

Domain is the set of allowed inputs; range is the set of possible outputs.

Memory trick: Domain in, range out.

One-one vs onto

“A bijection (both) is a perfect pairing with no leftovers.”

A one-one (injective) function never repeats outputs; an onto (surjective) function hits every output.

Memory trick: Injective: no repeats; surjective: no gaps.

Composite function

“Functions can be chained like an assembly line, each stage feeding the next.”

Applying one function to the result of another, (f∘g)(x) = f(g(x)).

(f∘g)(x)=f(g(x))

Memory trick: Work inside-out.

Discriminant

“Check the discriminant and you know the roots' nature before solving.”

For ax²+bx+c=0, D = b²−4ac decides the roots: D>0 two real, D=0 equal, D<0 complex.

D = b²−4ac

Memory trick: D<0 → complex roots (not 'none').

Quadratic formula

“One formula unlocks every quadratic ever written.”

The roots of ax²+bx+c=0 are x = [−b ± √(b²−4ac)]/2a.

x=(−b±√D)/2a

Memory trick: Memorise it cold.

Sum and product of roots

“You can build a quadratic straight from its roots — no expansion needed.”

For ax²+bx+c=0, roots sum to −b/a and multiply to c/a.

Memory trick: Sum −b/a, product c/a.

Imaginary unit

“'Imaginary' numbers run the very real electronics in your phone.”

i is defined by i² = −1, extending numbers beyond the real line.

i² = −1

Memory trick: i² = −1.

Modulus of a complex number

“A complex number is just a point on a plane; its modulus is how far out it sits.”

|z| = √(a²+b²) is z's distance from the origin.

|z|=√(a²+b²)

Memory trick: Distance from origin.

Complex conjugate

“Multiplying by the conjugate is the trick to clear i from a denominator.”

The conjugate of a+bi is a−bi; z times its conjugate is |z|² (real).

z·z̄=|z|²

Memory trick: Flip the sign of i.

Arithmetic progression

“Stacking equal steps — like saving a fixed amount every month.”

A sequence with a constant common difference; nth term a+(n−1)d.

aₙ=a+(n−1)d

Memory trick: Add a fixed step each time.

Geometric progression

“Compound interest and viral sharing both grow geometrically.”

A sequence with a constant ratio; nth term a·r^(n−1).

aₙ=a·r^(n−1)

Memory trick: Multiply by a fixed ratio.

Sum of infinite GP

“Add ½ + ¼ + ⅛ + … forever and you get exactly 1.”

Converges to a/(1−r) only when |r| < 1.

S∞=a/(1−r)

Memory trick: Needs |r| < 1.

AM ≥ GM inequality

“A powerful shortcut for finding minimum or maximum values.”

The arithmetic mean of positive numbers is at least their geometric mean.

Memory trick: AM ≥ GM, equal only when all equal.

Fundamental counting principle

“3 shirts and 4 trousers make 12 outfits — multiplication, not addition.”

If one task has m ways and another n ways, together they have m×n ways.

Memory trick: 'And' multiplies; 'or' adds.

Permutation

“The number of ways to award gold, silver and bronze — order matters.”

An ordered arrangement of r from n: nPr = n!/(n−r)!.

nPr=n!/(n−r)!

Memory trick: Order matters → permutation.

Combination

“Choosing a committee — the order you pick people doesn't matter.”

An unordered selection of r from n: nCr = n!/[r!(n−r)!].

nCr=n!/(r!(n−r)!)

Memory trick: Order irrelevant → combination.

Binomial theorem

“Pascal's triangle hands you the coefficients row by row.”

(a+b)^n expands with coefficients nCr, powers of a and b summing to n.

Tᵣ₊₁=nCr·a^(n−r)b^r

Memory trick: Exponents add to n each term.

General term of a binomial

“To find a specific term, set the exponent and solve for r.”

The (r+1)th term is nCr·a^(n−r)·b^r.

Tᵣ₊₁

Memory trick: Mind the +1 in the index.

Pythagorean identity

“The unit circle in one equation.”

For any angle, sin²θ + cos²θ = 1.

sin²θ+cos²θ=1

Memory trick: It's the squares that sum to 1.

Radian measure

“Radians tie an angle to the arc length it sweeps, which is why calculus loves them.”

π radians = 180°; calculus formulas assume radians.

π rad = 180°

Memory trick: Switch to radians before differentiating.

Range of sine and cosine

“If your working gives sinθ = 1.4, you've slipped somewhere.”

sinθ and cosθ always lie between −1 and 1.

Memory trick: Never outside [−1, 1].

Sine and cosine addition

“These unlock angles you can't read straight off the triangle.”

sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB.

Memory trick: Watch the sign flip in cosine.

Sine rule

“It cracks triangles when you know an angle and its opposite side.”

In any triangle, a/sinA = b/sinB = c/sinC.

a/sinA=b/sinB

Memory trick: Pair each side with its opposite angle.