HSC Maths Ext 1 key terms

A rule that pairs each element of a domain with exactly one element of a codomain. Different inputs may share an output, but no single input may produce two outputs.

The complete set of input values for which a function is defined. For a formula, it excludes values causing division by zero or a negative under a square root.

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

A function where distinct inputs always give distinct outputs, so no two domain elements share an output value. Such functions pass both the vertical and horizontal line tests.

A graphical check: if every horizontal line meets a curve at most once, the curve represents a one-to-one function on that interval and therefore has an inverse.

The function that reverses another, sending each output back to its unique input. It exists only when the original is one-to-one; its graph is the reflection in the line y = x.

Reducing a function's domain, often to an interval where it becomes one-to-one, so that an inverse can be defined. The rule stays the same while the allowed inputs shrink.

A function satisfying f(-x) = -f(x) for every x in its domain. Its graph has rotational symmetry about the origin, as with sine and cubing.

A function satisfying f(-x) = f(x) for every x in its domain. Its graph is symmetric about the y-axis, as with cosine and squaring.

The non-negative distance of a number from zero, written |x|. It equals x when x is non-negative and -x when x is negative, producing a V-shaped graph.

An angle measure based on arc length: one radian subtends an arc equal to the circle's radius. A full turn is 2π radians, so degrees convert by multiplying by π/180.

The circle of radius one centred at the origin. A point at angle θ on it has coordinates (cos θ, sin θ), which defines the trigonometric functions for all angles.

The trigonometric function giving, on the unit circle, the y-coordinate of the point at angle θ. Its graph oscillates between -1 and 1 with period 2π.

The trigonometric function giving, on the unit circle, the x-coordinate of the point at angle θ. Its graph oscillates between -1 and 1 with period 2π and is even.

The ratio sin θ divided by cos θ, equal to the gradient of the radius line's perpendicular. It is undefined where cosine is zero and repeats every π radians.

Half the vertical distance between the maximum and minimum values of an oscillating function. For y = a sin(bx) + c the amplitude is |a|.

The smallest positive input interval after which a repeating function reproduces its values. For y = a sin(bx) the period is 2π divided by |b|.

The horizontal translation of a trigonometric graph, found from the constant inside the bracket. A positive shift moves the curve left when written as x + c.

An equation true for every angle, such as sin²θ + cos²θ = 1. These are used to simplify expressions, prove other results, and transform integrals.

Identities expressing a trigonometric function of 2θ in terms of θ, for example sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ - sin²θ. Useful for integration and solving.

The distance along a curve. On a circle of radius r, an arc subtending angle θ radians has length rθ; for a full circle this gives 2πr.

The area of the region bounded by two radii and an arc. For radius r and angle θ in radians it equals half of r²θ.

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

The value a function's output approaches as the input approaches a chosen point. Limits underpin derivatives and are taken from both sides where relevant.

A rule for differentiating a composite function: differentiate the outer function, then multiply by the derivative of the inner function. In symbols, d/dx f(g(x)) = f'(g(x))g'(x).

A rule for differentiating a product: the derivative of uv equals u'v + uv'. Each factor is differentiated in turn while the other is held fixed.

A rule for differentiating a fraction: the derivative of u/v equals (u'v - uv') divided by v². The numerator order matters and v must be nonzero.

The straight line touching a curve at a point with the same gradient as the curve there. Its gradient equals the derivative evaluated at that point.

The line perpendicular to the tangent at a point on a curve. Its gradient is the negative reciprocal of the tangent's gradient.

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

The derivative of the derivative, measuring how the gradient itself changes. Its sign helps classify stationary points as maxima or minima.

A function whose derivative is the given function. Any two antiderivatives differ by a constant, so indefinite integrals include an arbitrary constant of integration.

The limit of sums of signed rectangular areas under a curve between two limits. It gives net area, counting regions below the axis as negative.

The region enclosed by two graphs, found by integrating the upper function minus the lower function between their intersection points. Intersections must be located first.

An ordered arrangement of objects selected from a group. The number of ways to arrange r objects chosen from n distinct objects is n! divided by (n - r)!.

A selection of objects where order does not matter. The number of ways to choose r objects from n distinct objects is n! divided by r!(n - r)!.

The product of all positive integers up to a given number, written n!. By convention 0! equals 1, which keeps counting formulas consistent.

If one choice can be made in m ways and a second in n ways, the pair can be made in m times n ways. It extends to any number of successive stages.

If more items than containers are distributed, at least one container holds two or more. It proves existence results without constructing an explicit example.

A proof that assumes the statement is false, then derives a logical impossibility. Concluding the assumption fails establishes the original statement.

A proof for statements about all positive integers: establish a base case, assume the statement for some integer, then show it follows for the next. This proves it for all.

The initial instance verified at the start of an inductive proof, usually for the smallest integer in scope. Without it the induction chain has nothing to start from.

A single case where a statement fails, which is enough to disprove a universal claim. Finding one is a valid disproof but never proves a statement true.

A quantity with both magnitude and direction, often written as a column or as components along axes. Vectors add tip-to-tail and can be scaled by numbers.

The length of a vector, found from its components using Pythagoras. For a vector with components a and b it equals the square root of a² + b².

A vector of length one, often used to specify direction. Any nonzero vector can be divided by its magnitude to produce a unit vector in the same direction.

A scalar formed from two vectors by multiplying matching components and adding. It equals the product of magnitudes times the cosine of the angle between them.

The component of one vector along the direction of another, found using the dot product. It gives the shadow of a vector on a line.

The arithmetic average, found by adding all values and dividing by how many there are. It is sensitive to extreme values, unlike the median.

A measure of how spread out values are around the mean, equal to the square root of the variance. Larger values indicate data further from the average.

The average of the squared deviations from the mean. Squaring makes it non-negative but in squared units, so its square root gives the standard deviation.

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

The number of standard deviations a value lies from the mean, found by subtracting the mean and dividing by the standard deviation. It allows comparison across distributions.