Glossary

Every term, in one place

The same 18 definitions the lessons link to inline, wherever a sentence uses them — gathered here so a definition is never more than one page away.

A

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°).

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.
completing the square: Rewriting a quadratic ax² + bx + c as a(x − p)² + q by building the perfect square matching its x² and x terms, then correcting the constant. Reads off a turning point, a range, or — for x² + y² + 2gx + 2fy + c = 0 — a circle's centre and radius directly, with the coordinate always the NEGATIVE of the value inside the bracket.
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.
counter-exampledisproof by counter-example, counterexample: A single case that fails a 'for all' claim, which is enough on its own to disprove it. A claim true for every case in a set is really one AND-statement per case, and one false part makes the whole AND false — which is why disproof never needs more than one case, however large or unbounded the set is.

discriminantb² − 4ac: The quantity b² − 4ac for a quadratic ax² + bx + c, whose sign decides how many real roots it has — positive for two, zero for one repeated, negative for none. Checked whenever a quadratic newly appears (from a substitution, a division, or a factorisation search) before spending time hunting for real roots that may not exist.
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.

equation of a circlecentre-radius form, general form (circle): A circle of centre (a, b) and radius r satisfies (x − a)² + (y − b)² = r² — Pythagoras' theorem applied to every point at distance r from the centre. Usually given multiplied out as x² + y² + 2gx + 2fy + c = 0 ('general form'), which has to be converted back by completing the square before a centre or radius can be read off at all.
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.

identity≡: An equation true for every value of the variable, marked with ≡ rather than = — licensing substitution of ANY value to derive a further result, unlike an ordinary equation, which is only true for the specific value(s) being solved for. Every trig identity, the division algorithm, and every double-angle or compound-angle formula are identities in exactly this sense.

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.

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.

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.

point of inflectioninflexion point, stationary point of inflection: A point where a curve's concavity changes — where d²y/dx² changes sign — which does not require the gradient to be zero there at all. The special case where a genuine stationary point (dy/dx = 0) also happens to be one is exactly what makes the second derivative test return zero: the gradient's own rate of change is momentarily zero without deciding whether it was increasing or decreasing.

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.

stationary pointturning point: A point on a curve where dy/dx = 0 — the gradient momentarily flat, found by differentiating, setting the result to zero, and solving; the matching y-coordinate then comes from substituting back into the ORIGINAL function, never into dy/dx.

tangenttangent line, tangent to a curve: A straight line that touches a curve at one point without crossing into it there. For a circle, a tangent always meets its radius at a right angle, so its gradient is the negative reciprocal of the radius's gradient; for a general curve, the tangent at a point shares that point's gradient, found by differentiation.

unit circle: A circle of radius 1 centred at the origin, whose point at angle θ (measured anticlockwise from the positive x-axis) is, by definition, (cos θ, sin θ) — the extension of sine and cosine to any real angle, and the single picture every trig identity and every solution-generation rule in this course reduces to.