Glossary

Every term, in one place

The same 28 definitions the lessons link to inline, wherever a sentence uses them — gathered here so a definition is never more than one page away.

A

algebraic long divisionpolynomial long division: Dividing one polynomial by another using the divide–multiply–subtract–bring-down layout of ordinary long division, producing a quotient and a remainder of lower degree than the divisor — fully creditable alongside comparing coefficients, though every subtraction is a fresh place for a sign error to enter.
antiderivativeindefinite integral: A function F(x) whose derivative is a given f(x), i.e. F'(x) = f(x) — found by reversing a standard differentiation result term by term. Every antiderivative of the same f(x) differs only by a constant, which is exactly why that constant cancels out of a definite integral's subtraction.
asymptotehorizontal asymptote, vertical asymptote: A line a curve approaches arbitrarily closely without ever touching or crossing it — a horizontal asymptote where a function settles as x → ±∞ (as eˣ does at y = 0), or a vertical one where a function diverges because a denominator or an argument hits zero (as ln x does at x = 0, or tan θ does at θ = 90°).

C

chain ruledy/dx = (dy/du)(du/dx): The rule for differentiating a composite function, dy/dx = (dy/du)·(du/dx) — differentiate the outer function with respect to whatever was fed into it, then multiply by the derivative of that inner function itself. Also the mechanism behind dy/dx = 1/(dx/dy): the identity that a function and its own inverse relationship undo each other, differentiated.
composite functionfg(x), composition of functions: fg(x) means f(g(x)) — apply g first, then feed its output into f. A genuinely different function from gf(x) in general, and a composite built from a domain-restricted inner or outer function inherits a domain restriction of its own, translated through whichever function acts first.
compound angle formulasin(A±B), cos(A±B), tan(A±B): sin(A±B) ≡ sinAcosB ± cosAsinB, cos(A±B) ≡ cosAcosB ∓ sinAsinB, and tan(A±B) ≡ (tanA±tanB)/(1∓tanAtanB) — printed in the exam formula booklet, and the identity every double-angle formula and every harmonic-form derivation is built from by substituting B = A or by expanding and comparing coefficients.
continuitycontinuous function: A curve having no breaks, jumps, or asymptotes across an interval — the second, load-bearing condition (alongside a sign change) for proving a root exists between two points, since a sign change caused by a discontinuity (like 1/x crossing x = 0) proves nothing at all.

D

division algorithm: The identity f(x) ≡ D(x)Q(x) + R(x) that dividing any polynomial f(x) by a divisor D(x) always satisfies, with R(x)'s degree strictly less than D(x)'s — a division is finished exactly when what's left has dropped below the divisor's own degree.
domain: The set of inputs a function is actually applied to. Genuinely independent of the range: a domain can be completely unrestricted while the range is still bounded, because a quadratic's own turning point — not its domain — is what limits its outputs.
double angle formulasin 2A, cos 2A, tan 2A: sin 2A ≡ 2 sin A cos A, cos 2A ≡ cos²A − sin²A ≡ 2cos²A − 1 ≡ 1 − 2sin²A, and tan 2A ≡ 2 tan A/(1 − tan²A) — each one the compound angle formula with B set equal to A, not a separate fact to memorise. Which of cos 2A's three equivalent forms to use is decided by matching it to whatever function the rest of an equation is already written in.

E

even function: A function g for which g(−x) = g(x) for every x in its domain — graphically symmetric about the y-axis. y = f(|x|) is even for every choice of f, with no exceptions, because |−x| = |x| makes the mirroring automatic before any specific f is even chosen.
exponential growth and decaygrowth and decay model: A quantity modelled as N = N₀aᵗ (or N₀eᵏᵗ), growing for a > 1 and decaying for 0 < a < 1, where 'initial' means specifically the value at t = 0. A rate-of-change question needs the model DIFFERENTIATED first, then a value substituted — substituting into the model itself answers a different question and earns nothing.

F

factor theorem: For a polynomial f(x), (ax − b) is a factor of f(x) exactly when f(b/a) = 0 — the zero-remainder case of the remainder theorem, used to hunt for factors by substitution before dividing at all.

I

inverse functionf⁻¹(x): The function that undoes f, found by swapping x and y in y = f(x) and rearranging — exists only where f is one-one, since a many-one f would need its inverse to send one output back to two different inputs. Its graph is the reflection of f's graph in the line y = x, and f⁻¹f(x) = ff⁻¹(x) = x for every x in the appropriate domain.
inverse trigonometric functionsarcsin, arccos, arctan: arcsin, arccos and arctan invert sin, cos and tan restricted to the one interval around θ = 0 on which each is one-one — arcsin: domain [−1,1], range −90° to 90°; arccos: domain [−1,1], range 0° to 180°; arctan: domain all reals, range the open interval −90° to 90°. A value outside the domain, such as arcsin(1.5), has no angle to invert and is undefined, not merely large.
iterationiterative method, fixed-point iteration: Rearranging f(x) = 0 into x = g(x) and repeatedly computing xₙ₊₁ = g(xₙ) from a starting value until the sequence settles — the value it settles on, L, satisfies L = g(L), algebraically the same equation as the original f(x) = 0, so the process is guaranteed to be homing in on a genuine root, not an arbitrary steady number.

L

log-linear graphlogarithmic graph, log-log graph: A graph of logged variables used to turn a curve of unknown shape into a straight line whose gradient and intercept hand you a model's own constants — plotting log y against log x straightens y = axⁿ (gradient n, intercept log a); plotting log y against unlogged x straightens y = kbˣ (gradient log b, intercept log k). The gradient never needs un-logging; the intercept always does.

M

many-one function: A function in which at least one output is shared by more than one input — e.g. f(x) = x² on all of ℝ, since f(−3) = f(3). Has no inverse function on its full domain; an inverse only exists once the domain is restricted down to a one-one piece.
modulus function|x|, absolute value: |x| = x when x ≥ 0, and |x| = −x when x < 0 — never negative. y = |f(x)| reflects only the parts of f that dipped below the x-axis back up above it; y = f(|x|) keeps the x ≥ 0 half of f unchanged and discards the x < 0 half entirely, rebuilding it as a mirror image of the positive side — two genuinely different curves from the same starting f(x).

N

natural logarithmln x, ln: logₑx, written ln x — the inverse function of eˣ, with domain x > 0 and range all real numbers, defined precisely because eˣ is strictly increasing (hence one-one) across its whole domain. ln(eˣ) = x and e^(ln x) = x are simply what 'inverse' means, applied to this pair.

O

one-one functioninjective, one-to-one function: A function in which no two different inputs ever produce the same output — the condition a function must satisfy on a given domain before an inverse function can exist there, since an inverse would otherwise need to send one output back to two different inputs at once.

P

product rule(uv)' = u'v + uv': The rule for differentiating a product of two functions of x, (uv)' = u'v + uv' — both factors are changing as x changes, so the derivative has to account for each factor's own rate of change scaled by the other factor's current size, not the product of the two derivatives.

Q

quotient rule(u/v)' = (u'v − uv')/v²: The rule for differentiating one function divided by another, (u/v)' = (u'v − uv')/v² — not an independent fourth rule but the product rule and the chain rule applied to u multiplied by v⁻¹, which is exactly why rewriting a quotient as a product with a negative index is always a genuinely valid second route to the same derivative.

R

rangerange of a function: The set of outputs a function actually produces, given its domain and rule — not a set chosen freely, but a consequence of the two together. Swapping x and y to find an inverse function exchanges these roles exactly: the domain of f⁻¹ is the range of f, and vice versa.
rational expression: One polynomial divided by another. Simplifying one means factorising numerator and denominator fully and cancelling a genuine shared FACTOR — never a term that is merely added, since only a true factor multiplies the whole of both top and bottom.
reciprocal trigonometric functionssecant, sec, cosecant, cosec, csc, cotangent, cot: sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ = cos θ/sin θ — nothing more than reciprocals, with a well-documented naming trap (cosec pairs with SINE, not cosine, despite sharing a first syllable with 'cos'). sec θ and cosec θ can never take a value strictly between −1 and 1, since they reciprocate a function already bounded within it; cot θ, reciprocating unbounded tan θ, can take any real value.
remainder theorem: The remainder when a polynomial f(x) is divided by (ax − b) equals f(b/a) — found by substituting the one value that makes the divisor zero, with no division needed, because the division identity f(x) ≡ (ax−b)q(x) + r collapses to r there.

S

sign-change rulechange-of-sign method, location of roots: If f(a) and f(b) have opposite signs and f is continuous on [a,b], then f(x) = 0 has at least one root strictly between a and b — a genuinely different job from finding that root's decimal value, and one that proves AT LEAST one root exists, never exactly one. The same test, applied to a narrow interval half a unit either side of a claimed decimal place, proves a value correct to that precision.