Math 30-1 key terms

A set of ordered pairs, or a correspondence between two sets, where each input may be linked to one or more outputs.

A relation in which every input value has exactly one output value. Its graph passes the vertical line test and can be evaluated consistently.

The set of all input values for which a relation or function is defined. Restrictions often come from denominators, even roots, or logarithms.

The set of all output values a relation or function can produce. It is found from the graph or by analyzing the rule.

A graphical check for functions: if any vertical line crosses a graph more than once, then one input has multiple outputs, so the relation is not a function.

A function where different inputs always produce different outputs. Its graph passes both the vertical and horizontal line tests, so its inverse is also a function.

A function that reverses another function's action: if f(a)=b, then the inverse maps b back to a. Its graph is a reflection of the original across y=x.

A graphical check for one-to-one functions: if every horizontal line crosses a graph at most once, then no two inputs share an output.

A new function formed by applying one function to the output of another, written (f∘g)(x)=f(g(x)); the inside function's range must fit the outside function's domain.

A change to a parent function's graph, such as a translation, reflection, stretch, or compression, often described by an equation with parameters.

A line that a graph approaches more and more closely as the input or output grows without bound, without necessarily touching or crossing it.

A function written as one polynomial divided by another, with the denominator not equal to zero; its graph often has holes or asymptotes.

A function that repeats its values in regular intervals, so f(x+p)=f(x) for a nonzero period p; trigonometric functions are common examples.

An angle placed in the coordinate plane with its vertex at the origin and its initial side along the positive x-axis; rotation determines its measure and quadrant.

The positive acute angle between the terminal side of an angle and the x-axis; it helps evaluate trigonometric ratios in any quadrant.

Angles in standard position that share the same terminal side; they differ by integer multiples of 360 degrees or 2π radians.

A unit of angle measure equal to the angle subtended at a circle's center by an arc whose length equals the radius; one full circle is 2π radians.

A circle of radius one centered at the origin, used to define sine and cosine as the y- and x-coordinates of points on the circle.

For a sinusoidal function, half the difference between its maximum and minimum values; it measures how far the graph varies from its midline.

The smallest positive interval over which a periodic function repeats its values; for sine and cosine it is 2π before transformations.

A horizontal translation of a trigonometric graph, showing how far and in which direction the wave is moved left or right.

An equation involving trigonometric functions that is true for all values in its domain, often used to simplify expressions or solve equations.

The identity sin²θ + cos²θ = 1, which follows from the unit circle and connects sine and cosine for every real angle.

A formula describing all angles or values that satisfy a trigonometric equation, using an integer parameter to account for repeating periods.

A function of the form f(x)=a·b^x, where the base b is positive and not one; it shows repeated multiplication and has constant percentage change.

The exponent to which a base must be raised to produce a given number; log_b(x)=y means b^y=x for positive b≠1 and x>0.

A logarithm with base e, often written ln(x); it is the inverse of the natural exponential function and is common in growth and decay.

A rule for rewriting a logarithm in another base: log_b(x)=log_c(x)/log_c(b), allowing calculations with available calculator bases, where c is any positive base not equal to one.

Rules for combining logarithms, including product, quotient, and power laws, such as log_b(xy)=log_b(x)+log_b(y), valid where all logarithms are defined.

An equation containing a logarithm of an unknown value; solving often requires rewriting in exponential form and checking for restrictions and extraneous solutions.

A value produced while solving an equation that does not satisfy the original equation; it must be rejected, often because it makes a logarithm or denominator undefined.

An equation in which a variable appears in an exponent; solving may use matching bases, logarithms, or graphing to find all valid solutions.

If one choice can be made in m ways and another in n ways, then the two choices together can be made in m·n ways.

The product of all positive integers from 1 up to a given nonnegative integer n, written n!, with 0! defined as 1.

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 combinations of n distinct items taken r at a time is n!/[r!(n-r)!].

A formula that expands (a+b)^n into a sum of terms using binomial coefficients, with term r+1 equal to C(n,r)a^(n-r)b^r for nonnegative integer n.

The number C(n,r)=n!/[r!(n-r)!], read n choose r; it counts r-item subsets of an n-item set and appears in binomial expansions.

A triangular array where each entry is the sum of the two above it; its rows give binomial coefficients for expanding powers of a binomial.

In a binomial expansion, the term containing a particular power of one variable, often written as C(n,r)a^(n-r)b^r for (a+b)^n, where r runs from 0 to n.