STPM Mathematics 954 key terms

A rule that assigns to each element of a domain exactly one element of a codomain. Written f: X → Y, where f(x) denotes the unique output for input x.

The complete set of input values for which a function is defined. For a rational function, values making the denominator zero are excluded from the domain.

The set of all actual output values a function produces as its input varies over the domain. It is a subset of the codomain.

A function formed by applying one function to the result of another, written f∘g with (f∘g)(x) = f(g(x)). The domain of g must lie within the domain of f.

A function that reverses another, so f⁻¹(f(x)) = x for every x in the domain. It exists only when the original function is one-to-one.

A function where distinct inputs always give distinct outputs. Its graph passes the horizontal line test, and it has an inverse function.

An expression of the form aₙxⁿ + ... + a₁x + a₀ with non-negative integer powers and real coefficients. Its degree is the highest power with a non-zero coefficient.

A polynomial P(x) has factor (x − a) exactly when P(a) = 0. Used to factorise polynomials and find their roots.

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). Dividing by (ax − b) gives remainder P(b/a).

Decomposing a rational expression into a sum of simpler fractions with linear or quadratic denominators. Useful for integration and binomial expansion of rational functions.

The function |x| giving the non-negative distance of x from zero: |x| = x for x ≥ 0 and |x| = −x for x < 0. Its graph is V-shaped.

A statement comparing two expressions using <, ≤, > or ≥. Solving it means finding all values satisfying the comparison, often expressed as an interval.

A function of the form f(x) = aˣ with a > 0 and a ≠ 1. Its graph always passes through (0, 1) and has the x-axis as a horizontal asymptote.

The inverse of an exponential: logₐ y = x means aˣ = y. Laws include log(xy) = log x + log y and log(xⁿ) = n log x.

A line that a curve approaches ever more closely without necessarily meeting. Vertical asymptotes occur where a rational function's denominator is zero.

The value a function approaches as its input approaches a particular point. Written lim(x→a) f(x) = L; the function need not equal L at x = a.

A function is continuous at a point when the limit there exists, equals the function's value, and the function is defined. Informally, the graph has no break.

The instantaneous rate of change of a function, defined as the limit of a difference quotient. Geometrically it gives the gradient of the tangent to the curve.

A function is differentiable at a point when its derivative exists there. Differentiability implies continuity, but continuity does not guarantee differentiability.

A rule for differentiating composite functions: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). Essential for functions such as sin(3x²).

A rule for differentiating a product: if y = uv then dy/dx = u(dv/dx) + v(du/dx), where u and v are functions of x.

A rule for differentiating a quotient: if y = u/v then dy/dx = [v(du/dx) − u(dv/dx)]/v². Useful for rational functions of x.

A point on a curve where the derivative equals zero. It may be a local maximum, local minimum, or point of inflection, classified using the second derivative.

The derivative of the derivative, written d²y/dx². It measures how the gradient changes and helps classify stationary points as maxima or minima.

A technique for evaluating integrals by changing the variable, reversing the chain rule. The limits must also be converted when the integral is definite.

The integral of a function between two limits, written ∫ₐᵇ f(x) dx. It represents the signed area between the curve and the x-axis.

The region between a curve y = f(x) and the x-axis from x = a to x = b, found by a definite integral. Regions below the axis count as negative.

A quantity with both magnitude and direction, often written as a column or in i, j, k components. It differs from a scalar, which has magnitude only.

A vector giving the location of a point relative to a fixed origin, usually written OA for point A. Used to express lines and planes in vector form.

A vector with magnitude exactly one, obtained by dividing a vector by its magnitude. The unit vectors along coordinate axes are i, j and k.

The dot product of vectors a and b, defined as |a||b|cos θ, giving a scalar. It is zero when the vectors are perpendicular.

A vector parallel to a line indicating its orientation. A line through point A with direction d has equation r = a + t d, where t is a scalar parameter.

A variable whose value is a numerical outcome of a random experiment. It may be discrete, taking countable values, or continuous, taking any value in an interval.

A description of the possible values of a random variable together with their probabilities. For a discrete variable it is given by a probability function summing to one.

The mean value of a random variable, written E(X). For a discrete variable it is the sum of each value multiplied by its probability.

A measure of spread defined as Var(X) = E(X²) − [E(X)]². It is the average of the squared deviations from the mean; its square root is the standard deviation.

The distribution of the number of successes in n independent trials, each with the same success probability p. Written X ~ B(n, p), with mean np and variance np(1 − p).

A continuous, symmetric bell-shaped distribution defined by its mean μ and variance σ². Written X ~ N(μ, σ²); probabilities are found by standardising to Z.

The normal distribution with mean zero and variance one, denoted Z ~ N(0, 1). Any normal variable is converted using Z = (X − μ)/σ before using tables.

The positive square root of the variance, expressed in the same units as the data. It measures the typical distance of observations from the mean.

The statement assumed true at the start of a significance test, usually claiming no effect or no difference. Denoted H₀ and tested against an alternative hypothesis.

The statement accepted when sample evidence is strong enough to reject the null hypothesis. It may be one-tailed or two-tailed, denoted H₁.

The probability, chosen before testing, of rejecting a true null hypothesis, commonly 0.05 or 0.01. It defines the critical region of the test.

A value calculated from sample data and compared with a critical value to decide on the null hypothesis. Examples include the sample mean standardised as a z-score.

The set of test statistic values that leads to rejecting the null hypothesis. Its total probability under the null hypothesis equals the significance level.

Rejecting a null hypothesis that is actually true. Its probability equals the significance level chosen for the test.

Failing to reject a null hypothesis that is actually false. Its probability is denoted β, and 1 − β is called the power of the test.

A range of values, calculated from a sample, that is likely to contain an unknown population parameter. The confidence level states the long-run proportion of such intervals that capture the parameter.