PEI Math 521A key terms

A rule that assigns to each input value in its domain exactly one output value, often written f(x); passing the vertical line test distinguishes functions from general relations.

The domain is the set of all valid input values for which a function is defined; the range is the set of all output values the function actually produces.

A function that reverses another function, so f and g are inverses when f(g(x)) = x and g(f(x)) = x; it exists only where the original function is one-to-one.

A new function formed by applying one function to the output of another, written (f ∘ g)(x) = f(g(x)); order matters because composition is generally not commutative.

A function of the form f(x) = ax² + bx + c with a ≠ 0, whose graph is a parabola opening upward when a > 0 and downward when a < 0.

The highest or lowest point on a parabola, located at x = -b/(2a) for f(x) = ax² + bx + c; it represents the maximum or minimum value of the quadratic function.

The quantity b² - 4ac from the quadratic formula; its sign indicates whether a quadratic equation has two real roots, one repeated real root, or two complex conjugate roots.

A technique that rewrites a quadratic expression as a perfect square plus a constant, converting ax² + bx + c into vertex form and enabling solution of quadratic equations.

A function of the form f(x) = a·bˣ with b > 0 and b ≠ 1; it models constant multiplicative growth or decay and has a horizontal asymptote at y = 0.

The exponent to which a base must be raised to produce a given number: log_b(x) = y means bʸ = x. It is the inverse relationship of exponentiation.

A number of the form a + bi where a and b are real and i² = -1; complex numbers extend the real line so that every polynomial equation has a root.

Every nonconstant polynomial with complex coefficients has at least one complex root, so a polynomial of degree n has exactly n roots counted with multiplicity.

An expression of the form aₙxⁿ + ... + a₁x + a₀ with nonnegative integer exponents and real coefficients; its degree is the largest exponent with nonzero coefficient.

A function written as the ratio of two polynomials, P(x)/Q(x); it may have vertical asymptotes where Q(x) = 0 and horizontal or slant asymptotes determined by degrees.

A line that a curve approaches arbitrarily closely as the input or output grows without bound, without necessarily ever touching it; common types are vertical, horizontal, and slant.

A function in which distinct inputs always produce distinct outputs, so f(a) = f(b) implies a = b; such functions pass the horizontal line test and possess an inverse function.

A function such as sine, cosine, or tangent defined from ratios of sides in a right triangle or from coordinates on the unit circle; these functions are periodic.

The circle of radius 1 centered at the origin; a point at angle θ on it has coordinates (cos θ, sin θ), which defines the trigonometric functions for all real angles.

An angle measure defined as the arc length subtended on a unit circle; a full revolution is 2π radians, and degrees convert by multiplying by π/180.

An equation involving trigonometric functions that holds for all valid angles, such as sin²θ + cos²θ = 1 or the double-angle formulas used to simplify and prove expressions.

In any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C; used to solve non-right triangles.

A generalization of the Pythagorean theorem: c² = a² + b² - 2ab·cos C, relating all three sides of a triangle to one angle and used for oblique triangles.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².

The distance along a curve between two points; on a circle of radius r, an arc subtending central angle θ in radians has length s = rθ.

A curve obtained by intersecting a plane with a double cone: circle, ellipse, parabola, or hyperbola; each has a standard algebraic equation in two variables.

The value a function approaches as its input approaches a specified point; limits underpin continuity and derivatives and may exist even where the function is undefined.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient; geometrically it gives the slope of the tangent line.

A quantity representing accumulated change, computed as a limit of sums of areas; the definite integral gives signed area under a curve between two bounds.

It links differentiation and integration: the definite integral of a continuous function can be evaluated using any antiderivative, and differentiating an accumulation function recovers the integrand.

A property of a function at a point where the limit exists, the function is defined there, and the two are equal; informally, the graph has no break or hole.

A line touching a curve at a point with slope equal to the derivative there; its equation is y - f(a) = f'(a)(x - a).

The arithmetic average of a data set, found by summing all values and dividing by the count; sensitive to extreme values or outliers.

The middle value of an ordered data set, or the average of the two middle values when the count is even; resistant to outliers compared with the mean.

A measure of the typical distance of data values from the mean, equal to the square root of the variance; larger values indicate more spread.

A symmetric, bell-shaped probability distribution defined by its mean and standard deviation; roughly 68% of values fall within one standard deviation of the mean.

A number between 0 and 1 expressing how likely an event is, computed for equally likely outcomes as favorable outcomes divided by total outcomes.

The probability that event A occurs given that event B has occurred, written P(A|B) = P(A and B)/P(B), provided P(B) > 0.

Two events where the occurrence of one does not change the probability of the other; formally P(A and B) = P(A)·P(B).

An arrangement of items in which order matters; the number of permutations of n distinct items taken r at a time is n!/(n - r)!.

A selection of items in which order does not matter; the number of ways to choose r items from n is n!/(r!(n - r)!), written C(n, r).

The product of all positive integers up to a given positive integer n, written n!; by convention 0! = 1, and factorials count orderings of distinct objects.

A proof technique showing a statement holds for a base case and that if it holds for n then it holds for n + 1, establishing truth for all integers beyond the base.

A single specific case that satisfies the hypotheses of a statement but makes its conclusion false; one counterexample disproves a universal claim.

The statement formed by swapping the hypothesis and conclusion of a conditional; a true statement does not guarantee that its converse is true.

A sequence defined by one or more initial terms together with a rule expressing each later term in terms of earlier ones, such as the Fibonacci sequence.

A sequence in which consecutive terms differ by a constant called the common difference; the nth term is a₁ + (n - 1)d.

A sequence in which each term after the first is found by multiplying the previous term by a fixed nonzero constant ratio; the nth term is a₁·rⁿ⁻¹.

An optimization method that maximizes or minimizes a linear objective function subject to linear inequality constraints, with the optimum occurring at a vertex of the feasible region.

A set of two or more equations solved simultaneously; a solution must satisfy every equation, and the number of solutions depends on whether the graphs intersect, coincide, or are parallel.

How much a quantity changes relative to another, computed as the change in output divided by the change in input; the average rate over an interval is a difference quotient.