MTEL Mathematics (63) key terms

A rule that assigns to each input value in a specified domain exactly one output value in the codomain. It may be given by a formula, table, graph, or verbal description.

The complete set of input values for which a function or expression is defined. For a rational function, values making a denominator zero are excluded; for a radical, values making a radicand negative are excluded.

The set of all output values actually produced by a function as its input varies over the domain. It is a subset of the codomain and is found from the graph's vertical extent.

A function that reverses another function, so composing the two in either order returns the original input. It exists only when the original function is one-to-one; its graph is the reflection across the line y = x.

A function in which distinct inputs always yield distinct outputs, so no horizontal line meets its graph more than once. Such functions possess an inverse that is itself a function.

The operation forming f(g(x)), in which the output of the inner function g becomes the input of the outer function f. The domain excludes inputs where g(x) is undefined or outside f's domain.

A function of the form f(x) = ax² + bx + c with a ≠ 0, whose graph is a parabola. The vertex gives the maximum or minimum, and the discriminant b² − 4ac determines the number of real zeros.

The turning point of a parabola, located at x = −b/(2a) for f(x) = ax² + bx + c. It represents the function's minimum when a > 0 and its maximum when a < 0.

The vertical line through a parabola's vertex, x = −b/(2a), about which the graph is mirror-symmetric. Points equidistant from this line horizontally have equal function values.

The quantity b² − 4ac for a quadratic ax² + bx + c. Positive means two distinct real roots, zero means one repeated real root, and negative means two nonreal complex conjugate roots.

A function of the form f(x) = a·bˣ with b > 0 and b ≠ 1. It grows when b > 1 and decays when 0 < b < 1, and it models constant percentage change such as compound interest or population growth.

The exponent to which a base must be raised to produce a given number: log_b(x) = y means bʸ = x. The logarithm is the inverse of the exponential function with the same base.

A number of the form a + bi, where a and b are real and i² = −1. Complex numbers arise from square roots of negatives and allow every polynomial of positive degree to have a root.

For a complex number a + bi, the conjugate is a − bi, sharing the same real part and opposite imaginary part. Multiplying a complex number by its conjugate yields the real number a² + b².

Every nonconstant polynomial with complex coefficients has at least one complex root. Consequently, a degree-n polynomial factors completely into n linear factors over the complex numbers, counting multiplicity.

An expression of the form aₙxⁿ + … + a₁x + a₀ with nonnegative integer exponents and real coefficients. Its degree is the highest exponent with a nonzero coefficient.

An input value x for which f(x) = 0. Zeros correspond to the x-intercepts of the graph and to the roots of the associated equation.

A test for rational zeros of a polynomial with integer coefficients: any rational zero p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

A polynomial f(x) has a factor (x − c) exactly when f(c) = 0. It links polynomial division, zeros, and factorization, and it underlies the remainder theorem.

A graphical check that a function is one-to-one: if no horizontal line intersects the graph more than once, the function has an inverse function. Failure of the test means the inverse is not a function.

The measure of a line's steepness, calculated as the change in y divided by the change in x between two points. It equals the tangent of the line's angle of inclination.

A function of the form f(x) = mx + b, whose graph is a straight line with slope m and y-intercept b. It models constant rate of change.

Distinct lines in the same plane that never intersect. In the coordinate plane they have equal slopes, though vertical lines are parallel to other vertical lines.

Lines that intersect at a right angle. Nonvertical perpendicular lines have slopes that are negative reciprocals, so their product is −1; a horizontal line is perpendicular to a vertical line.

A sequence in which each term after the first is obtained by adding a fixed constant called the common difference. Its nth term is a₁ + (n − 1)d.

A sequence in which each term after the first is obtained by multiplying by a fixed nonzero constant called the common ratio. Its nth term is a₁rⁿ⁻¹.

The sum of the terms of a sequence. An arithmetic series sums to n(a₁ + aₙ)/2; a finite geometric series sums to a₁(1 − rⁿ)/(1 − r) when r ≠ 1.

A ratio of two sides of a right triangle relative to an acute angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

The circle of radius 1 centered at the origin. A point at angle θ on it has coordinates (cos θ, sin θ), which extends trigonometric functions to all real angles and reveals their periodicity.

An angle measure defined so that one radian subtends an arc equal in length to the circle's radius. A full revolution is 2π radians, so π radians equals 180 degrees.

For a triangle with sides a, b, c opposite angles A, B, C: a/sin A = b/sin B = c/sin C. It solves triangles when two angles and a side, or two sides and a nonincluded angle, are known.

The relation c² = a² + b² − 2ab·cos C for a triangle. It generalizes the Pythagorean theorem to nonright triangles and solves triangles given two sides and the included angle or all three sides.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². Its converse tests whether a triangle is right.

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 at a point.

The limit of sums of function values times interval widths, interpreted as signed area under a curve. The definite integral accumulates change, and the fundamental theorem links it to antiderivatives.

The value a function's output approaches as its input approaches a specified point. Limits underpin continuity, derivatives, and integrals; a function is continuous where its limit equals its value.

A number expressible as a fraction p/q of integers with q ≠ 0. Rational numbers have decimal expansions that terminate or eventually repeat, and they are closed under addition, subtraction, multiplication, and division by nonzero values.

A real number that cannot be written as a ratio of integers, such as √2 or π. Its decimal expansion is infinite and nonrepeating, and it is neither terminating nor eventually periodic.

Any number on the continuous number line, including all rational and irrational numbers. The real numbers are ordered and complete, meaning every nonempty set bounded above has a least upper bound.