JC Mathematics key terms

A rule that assigns to each element in a set exactly one element in another set. The set of inputs is the domain; the set of outputs is the range.

The complete set of input values for which a function is defined. For example, the domain of f(x)=1/x is all real numbers except x=0.

The set of all output values that a function actually produces. It is a subset of the codomain and depends on both the rule and the domain.

A function formed by applying one function to the result of another, written (f∘g)(x)=f(g(x)). The domain of f∘g excludes x where g(x) is not in the domain of f.

The function that reverses the effect of a one-to-one function f, denoted f⁻¹, satisfying f⁻¹(f(x))=x and f(f⁻¹(y))=y. Its graph is the reflection of f's graph in the line y=x.

A function where different inputs always give different outputs: if f(a)=f(b) then a=b. Only one-to-one functions have inverse functions defined on their range.

A line that a curve approaches arbitrarily closely as it extends to infinity or near a point where the function is undefined. Common types are vertical, horizontal, and oblique asymptotes.

A change to a graph produced by altering its equation, such as translation, reflection, or scaling. For example, y=f(x)+a translates the graph of y=f(x) vertically by a units.

A transformation that shifts a graph without rotating or resizing it. Replacing x by x−h shifts the graph h units horizontally; adding k to the function shifts it k units vertically.

A transformation that stretches or compresses a graph. Multiplying the function by a factor a>0 scales it vertically; replacing x by x/b scales it horizontally by factor b.

A transformation that flips a graph over a line. Replacing f(x) by −f(x) reflects it in the x-axis; replacing x by −x reflects it in the y-axis.

A sequence in which each term after the first is obtained by adding a constant common difference d. The nth term is a+(n−1)d, where a is the first term.

A sequence in which each term after the first is obtained by multiplying by a constant common ratio r. The nth term is arⁿ⁻¹, where a is the first term.

The sum of the terms of a sequence. A finite series has a last term; an infinite series continues indefinitely and may converge or diverge.

A proof technique for statements about positive integers: prove the base case (usually n=1), assume true for n=k, then prove true for n=k+1. This establishes truth for all positive integers.

A quantity with both magnitude and direction, often represented as a directed line segment or as an ordered list of components. In three dimensions a vector is written ⟨a,b,c⟩ or ai+bj+ck.

A scalar product of two vectors: a·b = |a||b|cosθ, also computed as the sum of componentwise products. It is zero exactly when the vectors are perpendicular.

A vector product defined for two vectors in three dimensions, a×b, perpendicular to both. Its magnitude is |a||b|sinθ, equal to the area of the parallelogram they span.

A number of the form a+bi where a and b are real and i²=−1. The real part is a, the imaginary part is b, and the number can be represented as a point in the complex plane.

The distance of a complex number z=a+bi from the origin, written |z| and equal to √(a²+b²). For a real number it is the absolute value.

The angle a complex number makes with the positive real axis in the complex plane, measured in radians. It is defined modulo 2π; the principal argument usually lies in (−π,π].

For a complex number z=a+bi, the conjugate is z̄=a−bi, found by changing the sign of the imaginary part. The product z z̄ equals |z|².

The instantaneous rate of change of a function, defined as the limit of the difference quotient as the interval shrinks to zero. Geometrically it gives the slope of the tangent line.

A rule for differentiating composite functions: if y=f(g(x)), then dy/dx = f′(g(x))·g′(x). It is essential for differentiating nested expressions.

A straight line that touches a curve at a point and has the same slope as the curve there. Its slope equals the derivative at that point.

The limit of a sum of areas of thin rectangles under a curve, giving the definite integral; or the antiderivative, giving the indefinite integral. The two are linked by the Fundamental Theorem of Calculus.

A function whose derivative is the given function. If F′(x)=f(x), then F is an antiderivative of f; all antiderivatives differ by a constant.

The theorem connecting differentiation and integration: if F is an antiderivative of continuous f, then the definite integral of f from a to b equals F(b)−F(a).

An equation involving a function and one or more of its derivatives. Its solution is a function, often containing arbitrary constants unless initial conditions are given.

A first-order differential equation that can be written as dy/dx = f(x)g(y). It is solved by separating variables and integrating both sides.

A differential equation together with a condition specifying the value of the solution at a particular point. The condition determines the constant of integration.

A number between 0 and 1 measuring how likely an event is, defined as the number of favourable outcomes divided by the total number of equally likely outcomes, or as a long-run relative frequency.

The probability of event A given that event B has occurred, written P(A|B) and equal to P(A∩B)/P(B) when P(B)>0. It adjusts the sample space to B.

Two events where the occurrence of one does not affect the probability of the other. Formally, P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A).

A variable whose value is a numerical outcome of a random experiment. A discrete random variable takes countable values; a continuous one takes values in an interval.

The long-run average value of a random variable, computed as the sum of each value multiplied by its probability for a discrete variable. Also called the mean.

A measure of how spread out a random variable's values are around the mean, equal to the expected value of the squared deviation from the mean. Its square root is the standard deviation.

An arrangement of items in a definite order. The number of permutations of r items chosen from n distinct items is n!/(n−r)!.

A selection of items where order does not matter. The number of combinations of r items chosen from n distinct items is n!/(r!(n−r)!), written C(n,r) or ⁿCᵣ.

The product of all positive integers up to a given positive integer n, written n!. By convention 0!=1. It counts the number of ways to arrange n distinct items.

A rectangular array of numbers arranged in rows and columns. An m×n matrix has m rows and n columns and can represent a linear transformation.

A scalar associated with a square matrix that indicates whether the matrix is invertible. A 2×2 matrix [[a,b],[c,d]] has determinant ad−bc; a zero determinant means no inverse.

For a square matrix A, the matrix A⁻¹ satisfying AA⁻¹=A⁻¹A=I, the identity matrix. It exists exactly when det(A)≠0 and is used to solve matrix equations.

A proof technique that assumes the statement is false and derives a logical impossibility, thereby showing the statement must be true. Often used to prove irrationality or infinitude.

A single specific case that shows a general statement is false. For example, x=2 disproves the claim that all even numbers are composite.