CSEC Additional Mathematics key terms

A rule that assigns to each input value in a stated set exactly one output value. Written f(x), it pairs domain elements with range elements so no input has two different outputs.

The complete set of input values for which a function is defined. For a fraction, exclude values making the denominator zero; for a square root, keep the expression under the root non-negative.

The set of all output values a function actually produces as the input runs through its domain. Found by analysing the function's behaviour or by using the inverse.

A function formed by applying one function to the result of another, written fg(x) or f(g(x)). The output of the inner function becomes the input of the outer one.

The function that reverses another, written f⁻¹, so that f⁻¹(f(x)) = x. It exists only when each output comes from exactly one input; its graph mirrors the original in the line y = x.

A function in which different inputs always give different outputs, so no two domain values share an output. Only such functions possess a true inverse that is itself a function.

A function of the form ax² + bx + c with a ≠ 0, whose graph is a parabola. Its turning point, intercepts and line of symmetry follow from completing the square or using the discriminant.

The quantity b² − 4ac from a quadratic ax² + bx + c. Its sign tells how many real roots exist: positive gives two, zero gives one repeated root, negative gives none.

Rewriting ax² + bx + c as a(x + p)² + q, which reveals the vertex of the parabola and the maximum or minimum value of the quadratic expression.

The turning point of a parabola, where the curve reaches its maximum or minimum. For y = a(x − h)² + k the vertex is (h, k).

The vertical line through a parabola's vertex that divides the curve into two mirror-image halves. For y = ax² + bx + c it has equation x = −b/(2a).

The values of the variable that make an equation true. Graphically these are the points where the curve crosses the horizontal axis.

A polynomial p(x) has (x − a) as a factor exactly when p(a) = 0. It converts factor questions into simple substitution and evaluation.

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). This gives the remainder without performing long division.

An expression made of terms of the form axⁿ where n is a whole number, added together. Its degree is the highest power present.

Splitting a single rational expression into a sum of simpler fractions with linear or repeated linear denominators. Useful before integrating or expanding such expressions.

A fraction whose numerator and denominator are polynomials, such as (2x + 1)/(x − 3). Simplifying usually means factorising and cancelling common factors.

A statement comparing two expressions using <, ≤, > or ≥. Solving it gives a range of values, often shown on a number line or as an interval.

The absolute value of a number, written |x|, giving its distance from zero and so always non-negative. Equations and inequalities involving it usually split into separate positive and negative cases.

A function of the form y = aˣ where the variable sits in the exponent and a is a positive constant. It grows or decays at a rate proportional to its current size.

The power to which a base must be raised to give a stated number: logₐ n = x means aˣ = n. Logarithms turn multiplication into addition and solve exponential equations.

A logarithm taken to base e, written ln x, where e ≈ 2.71828. It is the inverse of the exponential function eˣ and appears throughout calculus and growth problems.

Rules allowing logarithms to be combined or split: log(ab) = log a + log b, log(a/b) = log a − log b, and log(aⁿ) = n log a. They are the main tool for solving logarithmic equations.

A measure of how steep a line is, calculated as the change in y divided by the change in x between two points. Parallel lines share it; perpendicular lines multiply to −1.

A relationship between x and y whose graph is a straight line, often written y = mx + c where m is the gradient and c the y-intercept.

Two lines meeting at a right angle. Their gradients m₁ and m₂ satisfy m₁ × m₂ = −1, so one gradient is the negative reciprocal of the other.

A unit of angle measure defined so that a full turn is 2π radians. One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius.

The distance along a curved part of a circle's circumference. With angle θ in radians and radius r, it equals rθ.

The area of the region enclosed by two radii and an arc. With angle θ in radians and radius r, it equals ½r²θ.

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite the matching angle. Used when you know a side and its opposite angle plus one more part.

In any triangle, a² = b² + c² − 2bc cos A. It finds a side from two sides and the included angle, or an angle from all three sides.

The area of any triangle equals ½ab sin C, using two sides and the angle between them. It works without needing the height.

An equation true for every angle, such as sin²θ + cos²θ = 1 or tan θ = sin θ/cos θ. Identities are used to simplify expressions and solve trigonometric equations.

Expansions such as sin(A + B) = sin A cos B + cos A sin B, and the matching forms for cos(A ± B) and tan(A ± B). They express a combined angle in terms of single angles.

Special cases of the compound angle formulas with A = B, giving sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A. Useful for integrating and simplifying.

An equation involving sine, cosine or tangent that must be solved for angles. Because these functions repeat, all solutions within a stated interval must be listed.

The value a function's output approaches as the input approaches a stated point, whether or not the function is defined there. Limits underpin the definition of the derivative.

The instantaneous rate of change of a function, written dy/dx or f′(x). Geometrically it gives the gradient of the tangent to the curve at a point.

The process of finding a derivative. Standard rules cover powers, sums, products, quotients, trigonometric functions, exponentials and logarithms.

A rule for differentiating a function of a function: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). Essential for composite expressions.

A rule for differentiating a product: if y = uv then dy/dx = u(dv/dx) + v(du/dx). Each factor is differentiated in turn while the other is held fixed.

A rule for differentiating a fraction: if y = u/v then dy/dx = [v(du/dx) − u(dv/dx)]/v². The denominator is squared and the order of subtraction matters.

The tangent is the line touching a curve whose gradient equals the derivative at that point. The normal is perpendicular to the tangent there, so its gradient is the negative reciprocal.

A point on a curve where the derivative equals zero, so the tangent is horizontal. It may be a maximum, a minimum or a point of inflection.

The derivative of the derivative, written d²y/dx². Its sign at a stationary point distinguishes a maximum from a minimum.

The largest and smallest values a function takes, either locally near a turning point or globally over an interval. Located using stationary points and endpoint checks.

The reverse of differentiation, used to recover a function from its gradient or to accumulate quantities. Definite integration between limits gives an area.

An integral evaluated between two limits, written with lower and upper bounds. It produces a number rather than a family of functions.

The region between a curve and the x-axis over an interval, found by definite integration. Portions below the axis contribute negative values, so signs must be handled carefully.

How quickly one quantity changes with respect to another, given by a derivative. In applied problems it links time to distance, volume, cost or population.

The mathematics of motion, describing displacement, velocity and acceleration. Velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity.

Problems where two related quantities change with time and the chain rule links their rates. A typical example relates the rate of increase of a sphere's radius to the rate of increase of its volume.

Finding the best possible value of a quantity, such as the largest volume or the least cost, by forming a function and locating its stationary points.

A method of maximising or minimising a linear objective subject to linear inequalities. The optimum occurs at a vertex of the feasible region.

The set of points satisfying all the constraints of a linear programming problem at once. It is usually a polygon whose vertices are tested in the objective function.