CEMC key terms

A rule that assigns to each input value in a specified set exactly one output value. The set of allowed inputs is the domain, and the set of resulting outputs is the range.

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

The set of all output values a function actually produces as its input varies over the domain. It is a subset of the codomain and may be found by analyzing the function's behavior.

The operation forming f(g(x)) by applying g first and then f. The domain of the composition excludes inputs where g(x) is undefined or where g(x) lies outside the domain of f.

A function that undoes another, so that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x on suitable domains. It exists only when the original function is one-to-one.

A function where distinct inputs always give distinct outputs, so f(a) = f(b) implies a = b. Such functions pass the horizontal line test and possess an inverse.

The expression x = (−b ± √(b² − 4ac)) / (2a) giving the solutions of ax² + bx + c = 0 when a ≠ 0. It solves any quadratic equation, including those with no real roots.

The quantity b² − 4ac associated with the quadratic ax² + bx + c. Its sign tells whether the equation has two real roots, one repeated real root, or no real roots.

An expression formed from variables and constants using only addition, subtraction, and multiplication by nonnegative integer powers, such as 3x⁴ − 2x + 7. Its degree is the highest power with nonzero coefficient.

The statement that dividing a polynomial P(x) by (x − a) leaves a remainder equal to P(a). It allows remainders to be computed by direct substitution rather than long division.

The statement that (x − a) is a factor of a polynomial P(x) exactly when P(a) = 0. It links roots of a polynomial to its linear factors.

A function of the form f(x) = a·bˣ with b positive and not equal to 1. It grows or decays by a constant factor for each unit increase in x.

The exponent to which a fixed base must be raised to produce a given number: log_b(x) = y means bʸ = x. Logarithms reverse the action of exponential functions.

In any triangle, the ratio a/sin A equals b/sin B equals c/sin C, where each side is opposite the corresponding angle. It is used when two angles and a side, or two sides and a non-included angle, are known.

The relation c² = a² + b² − 2ab·cos C in any triangle, where C is the angle between sides a and b. It generalizes the Pythagorean theorem to non-right triangles.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It is the special case of the cosine law where the included angle is 90°.

A ratio of two sides of a right triangle, such as sine, cosine, or tangent of an acute angle. These ratios depend only on the angle, not on the particular triangle's size.

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

The distance measured along a circle's curved edge between two points. For a central angle θ measured in radians, arc length equals rθ where r is the radius.

The area of the region bounded by two radii and the arc between them. For a central angle θ in radians, the area equals ½r²θ.

Triangles whose corresponding angles are equal and whose corresponding side lengths are in a common ratio. Their areas compare as the square of that ratio.

The relationship where an integer a divides an integer b exactly, leaving zero remainder, written a | b. Divisibility rules allow quick checks such as testing whether a number is divisible by 3.

An integer greater than 1 whose only positive divisors are 1 and itself, such as 2, 3, 5, and 7. Every integer above 1 factors uniquely into primes.

The expression of an integer as a product of prime numbers, unique up to order by the fundamental theorem of arithmetic. For example, 84 = 2² · 3 · 7.

The largest positive integer dividing each of two or more given integers. It can be found by comparing prime factorizations and taking the lowest power of each shared prime.

The smallest positive integer divisible by each of two or more given integers. It can be found from prime factorizations by taking the highest power of each prime that appears.

A system where integers are compared by their remainders upon division by a fixed modulus, written a ≡ b (mod n). Two numbers are congruent when their difference is a multiple of n.

A list of numbers in which each term after the first is obtained by adding a fixed constant called the common difference. Its nth term is a + (n − 1)d.

A list of numbers in which each term after the first is obtained by multiplying by a fixed nonzero constant called the common ratio. Its nth term is a·r^(n−1).

The sum of the terms of an arithmetic sequence. For n terms with first term a and last term l, the sum equals n(a + l)/2.

The sum of the terms of a geometric sequence. A finite one sums to a(1 − rⁿ)/(1 − r) for r ≠ 1, and an infinite one converges to a/(1 − r) when |r| < 1.

The value that the terms of a sequence or the outputs of a function approach as the index or input approaches a specified point. A limit may exist even when the expression is undefined at that point.

The property of a sequence or series whose terms approach a finite limiting value or whose partial sums approach a finite total. Otherwise the sequence or series is said to diverge.

An arrangement of objects in which order matters. The number of ordered arrangements of r objects chosen from n distinct objects is n!/(n − r)!.

A selection of objects in which order does not matter. The number of ways to choose r objects from n distinct objects is n!/(r!(n − r)!), often written C(n, r).

The product of all positive integers up to a given positive integer n, written n!. By convention 0! = 1, and n! counts the orderings of n distinct objects.

A number between 0 and 1 measuring how likely an event is, computed in equally likely settings as favourable outcomes divided by total outcomes. Probabilities of all mutually exclusive outcomes sum to 1.

Two events where the occurrence of one does not change the probability of the other, so P(A and B) = P(A) · P(B). This multiplication rule fails for dependent events.

The probability that event A occurs given that event B has already occurred, written P(A | B) and equal to P(A and B)/P(B) when P(B) > 0.

The event consisting of all outcomes where a given event does not occur. Its probability equals 1 minus the probability of the original event.

The observation that placing more than n objects into n containers forces at least one container to hold two or more objects. It proves existence of a repeated case without constructing it.

Whether an integer is even or odd. Parity arguments use the fact that sums and products of even and odd numbers follow fixed patterns, often to show a configuration is impossible.

A reasoning method that assumes the opposite of the desired conclusion and derives a logical impossibility, thereby establishing the original statement. It is common in irrationality and infinitude arguments.

A quantity or property that does not change under the allowed operations of a problem. Finding an invariant often shows that a target configuration can never be reached.