Mathematics Methods Level 4 key terms

A rule that assigns to each input value in a specified set exactly one output value, often written f(x); the input set is the domain and the resulting outputs form the range.

The complete set of input values for which a function is defined, excluding values that cause division by zero or take the square root of a negative number.

The set of all output values a function actually produces as its input varies across the domain; found by analysing the function's behaviour and graph.

A function formed by applying one function to the output of another, written f(g(x)); its domain excludes inputs where g is undefined or f(g(x)) is undefined.

A function that reverses the action of another, so f and its inverse undo each other; it exists only when the original function is one-to-one on its domain.

A function in which every output value corresponds to exactly one input value, so no horizontal line meets its graph more than once and an inverse exists.

A change to a parent graph produced by shifting, reflecting, stretching, or compressing it, described by adjustments to the parameters in its equation.

A rigid shift of a graph horizontally, vertically, or both without changing its shape or orientation, caused by adding or subtracting constants in the equation.

A stretch or compression of a graph away from or toward an axis, produced by multiplying the function or its input by a scale factor greater or less than one.

A graphical check stating that a function is one-to-one if and only if no horizontal line intersects its graph at more than one point.

An expression formed from variables and constants using only addition, subtraction, and multiplication by non-negative integer powers, such as 3x cubed minus 2x plus 7.

The highest power of the variable appearing with a nonzero coefficient in a polynomial, which determines the graph's overall end behaviour and maximum number of roots.

A result stating that a polynomial has factor (x minus a) exactly when substituting a gives zero, linking factors of a polynomial to its zeros.

A result stating that dividing a polynomial by (x minus a) leaves a remainder equal to the polynomial evaluated at a, allowing quick remainder calculation.

The number of times a particular root appears as a factor of a polynomial; it controls whether the graph crosses or touches the axis at that root.

A function written as one polynomial divided by another; its graph often has vertical asymptotes where the denominator is zero and horizontal or oblique asymptotes.

A line that a curve approaches increasingly closely as the input or output grows without bound, without necessarily ever meeting it.

A technique that rewrites a rational expression as a sum of simpler fractions with linear or quadratic denominators, useful for integration and simplification.

A function of the form a to the power x with positive base a not equal to one, characterised by constant proportional growth or decay and a horizontal asymptote.

The exponent to which a fixed base must be raised to produce a given number; it is the inverse operation to exponentiation and solves equations like a to the x equals b.

A logarithm taken with base e, written ln x, whose derivative is one over x and which frequently appears in calculus and growth models.

The function e to the power x, where e is approximately 2.718; it is its own derivative and models continuous growth and decay.

A rule allowing a logarithm in one base to be rewritten using another base, typically as log b over log a, so calculators can evaluate any logarithm.

A unit of angle measure defined so that a full turn is two pi radians, equating the angle to the arc length divided by the radius of a circle.

The circle of radius one centred at the origin, on which a point at angle theta has coordinates cosine theta and sine theta, defining trigonometric functions for all angles.

An equation involving trigonometric functions that holds for every value where both sides are defined, such as sine squared plus cosine squared equals one.

An identity expressing a trigonometric function of twice an angle in terms of functions of the original angle, for example sine two theta equals two sine theta cosine theta.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient and equal to the slope of the tangent line.

The value a function's output approaches as its input approaches a particular point, whether or not the function is defined at that point.

A differentiation rule for composite functions stating that the derivative of f(g(x)) is f prime of g(x) multiplied by g prime of x.

A differentiation rule stating that the derivative of a product of two functions equals the first times the derivative of the second plus the second times the derivative of the first.

A differentiation rule for one function divided by another, giving the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.

A straight line touching a curve at a point with the same slope as the curve there, whose slope equals the derivative at that point.

A point on a curve where the derivative equals zero, marking a local maximum, local minimum, or horizontal point of inflection.

The derivative of the derivative, describing how the rate of change itself changes and used to classify stationary points through concavity.

The way a curve bends, described as concave up when the second derivative is positive and concave down when it is negative.

A function whose derivative is the given function; antiderivatives differ only by an additive constant and form the basis of indefinite integration.

The signed area between a curve and the horizontal axis over an interval, computed as the limit of sums of rectangular areas and evaluated using antiderivatives.

The theorem linking differentiation and integration, stating that the definite integral of a function over an interval equals the difference of an antiderivative evaluated at the endpoints.

A technique for integrating composite expressions by reversing the chain rule, replacing part of the integrand with a new variable and adjusting the differential.

The region enclosed between two graphs, found by integrating the difference of the upper and lower functions across the interval where they bound the region.

The distance measured along a curve; for a circle it equals radius times the angle in radians, and for a general curve it is computed with an integral.

A description assigning probabilities to each possible value or range of values of a random variable, either as a table, formula, or density curve.

A numerical quantity whose value depends on the outcome of a random process, classified as discrete when it takes separate values or continuous otherwise.

The long-run average value of a random variable, computed as the sum or integral of each value weighted by its probability.

A symmetric bell-shaped probability distribution described by its mean and standard deviation, in which roughly 68 percent of values lie within one standard deviation of the mean.

A measure of how spread out values are around the mean, equal to the square root of the variance and expressed in the same units as the data.

A simplified mathematical description of a real situation, built from variables, assumptions, and equations, used to explain behaviour or make predictions.

The process of finding the input that gives the largest or smallest value of a quantity, usually by setting the derivative to zero and checking the nature of stationary points.