CLEP College Mathematics key terms

A rule that assigns to each input value in a stated domain exactly one output value. Passing the vertical line test on a graph confirms this one-output property.

The complete set of input values for which a function or relation is defined. For a rational function, values making the denominator zero are excluded from it.

The set of all output values a function actually produces from its domain. Unlike the codomain, it contains only values that are attained.

The constant rate of change of a line, computed as the change in output divided by the change in input between two points. Horizontal lines have slope zero; vertical lines have undefined slope.

The output value where a graph crosses the vertical axis, occurring when the input equals zero. In a linear model it often represents a starting amount.

A function whose graph is a straight line, expressible as f(x) = mx + b with constant slope m. It models situations with a fixed rate of change.

A function of the form f(x) = a·b^x with positive base b not equal to 1. It shows repeated multiplication, so equal input increases produce equal output ratios.

A function of degree two, written f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola with a vertex marking the maximum or minimum output.

The turning point of a parabola, where the function reaches its maximum or minimum value. Its horizontal coordinate equals −b/(2a) for a quadratic in standard form.

A graphical check stating that a curve represents a function only if no vertical line meets it at more than one point. It directly tests the single-output requirement.

Combining two functions so the output of one becomes the input of the other, written (f∘g)(x) = f(g(x)). The inner function is evaluated first.

A function that reverses another function's effect, so f⁻¹ undoes f and their composition returns the original input. Its graph is the reflection of the original across the line y = x.

The product of all positive integers from 1 up to a given whole number n, written n!. By convention 0! equals 1, and it counts orderings of n distinct items.

An arrangement of items in which order matters. The number of ways to arrange r items chosen from n distinct items is n! divided by (n − r)!.

A selection of items in which order does not matter. The count of r-item selections from n distinct items is n! divided by r!(n − r)!.

Two events that cannot occur in the same trial, so their intersection is empty. For such events the probability of either one occurring is the sum of their separate probabilities.

Events whose outcomes do not influence each other, so the probability of both equals the product of their individual probabilities. Occurrence of one leaves the other's probability unchanged.

The probability that an event occurs given that another event has already occurred, written P(A|B) and computed as P(A and B) divided by P(B).

A formula giving the probability that at least one of two events occurs: P(A or B) = P(A) + P(B) − P(A and B). The subtracted term corrects for double-counting overlap.

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

The middle value of an ordered data set, or the average of the two middle values when the count is even. It resists distortion by outliers.

The value or values appearing most frequently in a data set. A set may have one mode, several modes, or none.

A measure of how far data values typically spread from the mean, equal to the square root of the variance. Larger values indicate greater dispersion.

The average of the squared deviations of data values from their mean. Squaring makes it express spread in squared units, and its square root is the standard deviation.

A value dividing ordered data into four parts, each holding about one quarter of the observations. The second quartile equals the median.

The spread of the middle half of a data set, computed as the third quartile minus the first quartile. It is used to identify outliers.

A symmetric bell-shaped distribution in which the mean, median, and mode coincide. Roughly 68% of values fall within one standard deviation of the mean and 95% within two.

Interest computed on both the original principal and previously accumulated interest, so the balance grows by a constant factor each period. The formula is A = P(1 + r/n)^(nt).

Interest computed only on the original principal, given by I = Prt where r is the annual rate and t is time in years. The balance grows linearly.

The original amount of money borrowed or invested, before any interest is added. It serves as the base for computing interest charges or earnings.

The amount that a future sum of money is worth today when discounted at a given rate. It lets you compare payments occurring at different times.

The amount an investment or deposit will be worth at a later date after interest accumulates. It is the result of applying a growth factor to the present value.

Two angles whose measures add to 180 degrees. Angles on a straight line form such a pair.

Two angles whose measures add to 90 degrees. They commonly appear in right triangles, where the two acute angles are related this way.

The relationship a² + b² = c² holding for right triangles, where c is the hypotenuse. It allows finding a missing side length when two sides are known.

Triangles with equal corresponding angles and proportional corresponding side lengths. Their side ratios form a constant scale factor used to find unknown lengths.

A compound statement formed with 'and', true only when both component statements are true. It corresponds to the intersection of two sets.

A compound statement formed with 'and/or', true when at least one component statement is true. It corresponds to the union of two sets.

The set containing every element belonging to either of two sets, written A ∪ B. Elements appearing in both sets are listed only once.

The set of elements shared by two sets, written A ∩ B. It contains only elements present in both sets simultaneously.

The set of all elements in the universal set that are not in a given set, written A′. An element belongs to exactly one of a set or its complement.

A number expressible as a fraction of two integers with a nonzero denominator. Its decimal form either terminates or repeats indefinitely.

A real number that cannot be written as a fraction of two integers, so its decimal expansion never terminates or repeats. Familiar examples include the square root of 2 and pi.

The distance of a number from zero on the number line, always nonnegative. It is written with vertical bars and removes any negative sign.