Functions Terms

Function basics 16

• Average Rate of ChangeThe average rate of change of a function over an interval is the change in output divided by the change in input: (f(b) − f(a))/(b − a). It tells how fast the function changes, on average, from x = a to x = b. On a graph it is the slope of the line through the two endpoints.Algebra 1, Algebra 2, Precalculus
• Composition of FunctionsA composition of functions applies one function to the output of another. The composite f(g(x)), also written (f ∘ g)(x), means: find g(x) first, then put that result into f. If g(x) = x + 1 and f(x) = x^2, then f(g(2)) = f(3) = 9. Order matters.Algebra 2, Precalculus
• Dependent VariableA dependent variable is the output value that changes based on the input. It responds to changes in the independent variable: in y = 2x + 1, y is the dependent variable because its value depends on x. On a graph it goes on the vertical axis, because its value depends on what the input was.Grade 6, Grade 7, Grade 8
• DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Grade 8, Algebra 1, Algebra 2
• Domain and RangeDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Grade 8, Algebra 1, Algebra 2
• FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Grade 8, Algebra 1, Algebra 2
• Function NotationFunction notation is the way of writing a function as f(x), read "f of x": f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Grade 8, Algebra 1, Algebra 2
• Independent VariableAn independent variable is an input value that can be chosen or changed. The dependent variable changes based on it. In y = 3x, x is the independent variable and y is the dependent one. On a graph the independent variable goes on the horizontal axis; in an experiment it is the factor you change on purpose.Grade 6, Grade 7, Grade 8
• InputAn input is the value you put into a function or a rule. The rule acts on the input to produce the output: for f(x) = x + 2, the input 5 gives the output 7. For many functions, the input is the x-value. In an input-output table, the inputs fill the first column.Grade 6, Grade 7, Grade 8
• Inverse FunctionAn inverse function undoes another function: if f sends a to b, its inverse f^(−1) sends b back to a. For f(x) = 2x + 3, the inverse is f^(−1)(x) = (x − 3)/2. Only one-to-one functions have inverses, and the graph of f^(−1) is the mirror image of f across y = x.Algebra 2, Precalculus
• OutputAn output is the value a function or rule produces from a given input. You put a number in, apply the rule, and the result that comes out is the output. For y = 2x, the input 3 gives the output 6. In an input-output table the output sits in the second column.Grade 6, Grade 7, Grade 8
• Piecewise FunctionA piecewise function is a function defined by two or more formulas, each used on its own interval of the domain. To find an output, first check which interval the input belongs to, then use that piece's formula. Absolute value, step functions and tiered prices are common examples.Algebra 1, Algebra 2, Precalculus
• Range of a FunctionThe range of a function is the set of all output values (usually y-values) the function actually produces from its domain. For f(x) = x^2, every output is 0 or more, so the range is y ≥ 0. On a graph, the range is how far the graph reaches down and up.Grade 8, Algebra 1, Algebra 2
• RelationA relation is any set of ordered pairs that links inputs to outputs, such as {(1, 4), (2, 5), (2, 7)}. It can be shown as a list, table, mapping diagram or graph. A relation in which every input has exactly one output is a function; this one is not, because 2 has two outputs.Grade 8, Algebra 1
• Vertical Line TestThe vertical line test is a way to tell from a graph whether it shows a function: if any vertical line crosses the graph more than once, it is not a function. If every vertical line crosses it at most once, each input x has only one output y, so the graph is a function.Grade 8, Algebra 1, Algebra 2
• Zero of a FunctionA zero of a function is an input that makes the output 0: a number a with f(a) = 0. For f(x) = x − 5, the zero is 5. On a graph, the real zeros are the x-values where the curve meets the x-axis. You find zeros by solving f(x) = 0.Algebra 1, Algebra 2, Precalculus

Linear functions 2

• Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Grade 8, Algebra 1
• Rate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Grade 8, Algebra 1, Algebra 2

Polynomial functions 1

• End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x^3 + x rises on the left and falls on the right.Algebra 2, Precalculus

Exponential and logarithmic functions 8

• Compound InterestCompound interest is interest computed on the principal plus all the interest already earned, so the balance grows exponentially. The formula is A = P( 1 + r/n )^(nt), where P is the principal, r the annual rate, n the compoundings per year and t the years. It beats simple interest after the first period.Grade 7, Algebra 1, Algebra 2
• Euler's NumberEuler's number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1/n )^n gets closer and closer to as n grows. It is the base of natural exponential growth, e^x, and of the natural logarithm, natural log x.Algebra 2, Precalculus, Calculus
• Exponential DecayExponential decay is a decrease by the same percent in each equal time period, so the amount drops quickly at first and then more and more slowly. It is modeled by y = a(1 − r)^t, where a is the starting amount and r is the decay rate. The amount approaches 0 but never reaches it.Algebra 1, Algebra 2
• Exponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · b^x, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Algebra 1, Algebra 2, Precalculus
• Exponential GrowthExponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)^t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.Algebra 1, Algebra 2
• LogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. log base b of x = y means b^y = x, so log base 2 of 32 = 5 because 2^5 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Algebra 2, Precalculus
• Natural LogarithmThe natural logarithm of a positive number x, written natural log x, is the power you must raise e ≈ 2.718 to in order to get x. So natural log x = y means e^y = x. For example, natural log 1 = 0, natural log e = 1, and natural log 20 ≈ 3.0 because e^3 ≈ 20.1.Algebra 2, Precalculus, Calculus
• Properties of LogarithmsThe properties of logarithms are rules for rewriting logs: the log of a product is a sum, log base b of (MN) = log base b of M + log base b of N; the log of a quotient is a difference; and the log of a power is a multiple, log base b of (M^p) = p log base b of M. They come from the rules of exponents.Algebra 2, Precalculus

Sequences and series 8

• Arithmetic SequenceAn arithmetic sequence is a list of numbers in which you add the same amount each time to get the next term. That fixed amount is the common difference, d. In 4, 7, 10, 13, … the common difference is 3. The nth term is an = a1 + (n − 1)d.Grade 8, Algebra 1, Algebra 2
• Common DifferenceThe common difference is the fixed amount added to each term of an arithmetic sequence to get the next term, written d. Find it by subtracting any term from the term after it: in 7, 11, 15, 19, d = 11 − 7 = 4. A positive d makes the terms increase, a negative d makes them decrease.Grade 8, Algebra 1, Algebra 2
• Common RatioThe common ratio is the fixed number you multiply each term of a geometric sequence by to get the next term. Find it by dividing any term by the term before it: r = an/(an − 1). In 5, 20, 80, 320, the common ratio is 4. It can be a fraction or negative, but never 0.Algebra 1, Algebra 2
• Geometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · r^(n − 1). The terms grow when |r| > 1 and shrink when |r| < 1.Algebra 1, Algebra 2, Precalculus
• Recursive FormulaA recursive formula defines a sequence by giving its first term and a rule that builds each next term from the term or terms before it. For example, a1 = 5 and an = an − 1 + 4 make 5, 9, 13, 17. To find a later term, you compute every term before it, one step at a time.Algebra 1, Algebra 2
• SequenceA sequence is an ordered list of numbers that follows a rule, such as 3, 7, 11, 15, …. Each number is a term, and its position is 1st, 2nd, 3rd and so on. You can describe a sequence with a formula for the nth term, an, or by how each term comes from the one before it.Grade 8, Algebra 1, Algebra 2
• SeriesA series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written Sn. An infinite series has a sum only if its partial sums approach a fixed number.Algebra 2, Precalculus, Calculus
• Sigma NotationSigma notation is a compact way to write a long sum using the Greek capital letter Σ (sigma). In sum from k=1 to 4 of k^2, the index k starts at the number below sigma, stops at the number above it, and each value is put into the rule k^2: 1 + 4 + 9 + 16 = 30.Algebra 2, Precalculus, Calculus

Graphs 1

• GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Grade 5, Grade 6, Grade 7

Quadratic functions 3

• Axis of SymmetryThe axis of symmetry of a parabola is the vertical line through its vertex that splits it into two mirror-image halves. For y = ax^2 + bx + c, it is the line x = −b/2a. Points at equal distances on either side of this line have the same y-value.Algebra 1, Algebra 2
• ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x^2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Algebra 1, Algebra 2, Geometry
• Quadratic FunctionA quadratic function is a function that can be written f(x) = ax^2 + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola, which opens up when a > 0 and down when a < 0. Its highest or lowest point is the vertex.Algebra 1, Algebra 2, Precalculus

Rational and radical functions 3

• AsymptoteAn asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1/x has both: x = 0 and y = 0.Algebra 2, Precalculus, Calculus
• Rational ExpressionA rational expression is a fraction whose numerator and denominator are polynomials, such as (x + 3)/(x − 2) or 5/(x^2). The denominator cannot equal 0, so values that make it 0 are excluded. You simplify, add, subtract, multiply and divide rational expressions the same way you work with numerical fractions.Algebra 1, Algebra 2
• Rational FunctionA rational function is a function that can be written as a fraction of two polynomials, f(x) = (p(x))/(q(x)), where q(x) is not the zero polynomial. It is undefined wherever q(x) = 0, so its graph often has vertical asymptotes or holes. Examples: f(x) = 1/x and g(x) = (x + 2)/(x − 5).Algebra 2, Precalculus

Function transformations and families 2

• Parent FunctionA parent function is the simplest function of a family, before any shifts, stretches or reflections. For example, y = x^2 is the parent of every quadratic, and y = |x| is the parent of every absolute value function. Every other member of the family is a transformation of its parent's graph.Algebra 1, Algebra 2, Precalculus
• Transformation of FunctionsTransformation of functions is changing a function's rule so that its graph moves, flips or resizes. Adding k outside, f(x) + k, shifts the graph up; replacing x with x − h shifts it right; −f(x) reflects it over the x-axis; and a · f(x) stretches it vertically. The basic graph being changed is the parent function.Algebra 1, Algebra 2, Precalculus

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Updated November 2026·Reviewed by the Effortless Math team