Functions and function notation

You begin by deciding what counts as an input and whether it determines one output. Then you learn to read function notation and find values from tables, formulas, and graphs. You practice both directions: using an input to find an output, and finding every input that produces a stated output. You build the difference quotient, which measures average output change like average speed, then learn when outputs identify unique inputs. The two line tests and nine toolkit functions connect these ideas to pictures.

Lessons

Vocabulary

Relation rih-LAY-shun: A set of ordered pairs that connects inputs with outputs. It can have conflicting outputs and need not be a function. Like: A list of paired choices and results.
Ordered pair OR-derd pair: Two values written in a fixed order. Here the first is an input and the second is its paired output. Like: An address with a street location followed by a floor.
Set set: A collection of distinct members. Braces list them, and repeating a member does not create a new member. Like: A guest list with each guest counted once.
Element EL-uh-ment: One member of a set. Like: One guest on a guest list.
Natural numbers NACH-er-ul NUM-berz: The counting numbers 1, 2, 3 and so on, using this convention. Like: Counting one chair, two chairs, three chairs.
Integer IN-tuh-jer: A whole number or its negative, including zero. Fractions between whole numbers are not integers. Like: Whole steps left or right from a doorway.
Real number REE-ul NUM-ber: A number on the number line, including integers, fractions and numbers such as $square root of 2$ . Like: Any location along a straight sidewalk.
Variable VAIR-ee-uh-bul: A letter that represents a number or another chosen value. Like: A blank on a form waiting for a value.
Function FUNK-shun: A relation assigning exactly one output to every input in its domain. Like: A fixed-menu machine with one result per accepted choice.
Input IN-put: A value chosen for a function. The definition decides which values are allowed. Like: The button you choose on a vending machine.
Output OUT-put: The result a function assigns to a particular input. Like: The snack delivered after you choose a button.
Independent variable in-duh-PEN-dunt VAIR-ee-uh-bul: The input variable. You choose its value from the domain before the rule gives a result. Like: The age you choose before reading a growth model.
Dependent variable dih-PEN-dunt VAIR-ee-uh-bul: The output variable. Its value is determined by the chosen input and the rule. Like: A taxi fare determined by the trip distance.
Domain doh-MAYN: The set of all inputs accepted by the defined relation or function. Like: The list of buttons available on a machine.
Range raynj: The set of all outputs actually produced by the defined relation or function. Like: The snacks the machine can actually deliver.
Function notation FUNK-shun noh-TAY-shun: Writing an output as f(x), with the rule name outside and the input inside parentheses. Like: A recipe label followed by its ingredient amount.
Algebraic expression al-juh-BRAY-ik ek-SPRESH-un: A combination of values, letters and operations, without asserting an equality. Like: An unfinished arithmetic recipe.
Equation ee-KWAY-zhun: A statement that two expressions are equal, shown with an equals sign. Like: A balance scale with equal amounts on its two pans.
Formula FOR-myuh-luh: A mathematical instruction connecting quantities, often written as an equation. Like: A recipe giving exact operations.
Evaluate ih-VAL-yoo-ayt: Find the output at a given input by applying the rule or reading its representation. Like: Choose a button and see what arrives.
Solve solv: Find every allowed input that satisfies a stated equation or produces a specified output. Like: Find every button that delivers the snack you want.
Table of values TAY-bul uv VAL-yooz: A display pairing inputs with outputs. Each row or column keeps one pair together. Like: Paired cards with a choice and its result.
Tabular form TAB-yuh-ler form: A representation arranged as a table of input-output values. Like: Information organized into paired cards.
Algebraic form al-juh-BRAY-ik form: A formula giving the output directly in terms of the input. Here it is also called an explicit rule or explicit formula. Like: A written recipe rather than a lookup chart.
Implicit rule im-PLIS-it rool: An equation relating input and output without isolating the output. It defines a function only if each accepted input determines one output. Like: Instructions that identify a result without handing you a finished recipe.
Graph graf: A picture of ordered pairs at their coordinate locations. A graph of a function contains every pair (x, f(x)) for its allowed inputs. Like: A map of starting values and results.
Coordinate koh-OR-duh-nut: One number in a point's ordered address. Like: One part of a location on a city map.
x-coordinate eks koh-OR-duh-nut: The first number of a plotted pair, showing horizontal location and usually the input. Like: How far right or left an address lies.
y-coordinate why koh-OR-duh-nut: The second number of a plotted pair, showing vertical location and usually the output. Like: How far up or down an address lies.
Horizontal axis hor-ih-ZON-tul AK-sis: The left-right reference line on a graph, conventionally the input or x-axis. Like: The east-west line on a map.
Vertical axis VER-tih-kul AK-sis: The up-down reference line on a graph, conventionally the output or y-axis. Like: The north-south line on a map.
One-to-one function wun too wun FUNK-shun: A function where every output in its range comes from exactly one input. Different inputs always give different outputs. Like: Unique coat-check tickets paired with unique coats.
Vertical line test VER-tih-kul line test: A graph represents an output as a function of its horizontal input if every vertical line meets it at most once. Like: Hold the input still and count possible answers.
Horizontal line test hor-ih-ZON-tul line test: An established function is one-to-one if every horizontal line meets its graph at most once. Like: Hold the output still and count its possible starting inputs.
Toolkit functions TOOL-kit FUNK-shunz: The nine basic named functions that form this section's library of functions, used as building blocks for later formulas and graphs. Like: An alphabet of function shapes.
Constant KON-stunt: A value that stays fixed in the stated rule. Like: A flat fee that does not change with the trip.
Constant function KON-stunt FUNK-shun: A function f(x) = c that returns the same fixed real output for every real input. Like: A machine giving the same snack for every button.
Identity function eye-DEN-tih-tee FUNK-shun: The rule f(x) = x, returning the input unchanged. Like: A mirror that hands back the same number.
Absolute value AB-suh-loot VAL-yoo: A number's distance from zero, written between vertical bars. It is always nonnegative. Like: The distance home, regardless of walking direction.
Absolute value function AB-suh-loot VAL-yoo FUNK-shun: The rule f(x) = |x|, returning each input's distance from zero. Like: A machine that reports distance and drops direction.
Quadratic function kwah-DRAT-ik FUNK-shun: A function f(x) = a $x^{2}$ + bx + c with fixed real coefficients and a ≠ 0. Its toolkit form is f(x) = $x^{2}$ . Like: A square tile's area from its side length.
Cubic function KYOO-bik FUNK-shun: A function f(x) = a $x^{3}$ + b $x^{2}$ + cx + d with fixed real coefficients and a ≠ 0. Its toolkit form is f(x) = $x^{3}$ . Like: A cube's volume from its side length.
Reciprocal rih-SIP-ruh-kul: One divided by a nonzero number. A number times its reciprocal equals 1. Like: Turning a nonzero fraction upside down.
Reciprocal function rih-SIP-ruh-kul FUNK-shun: The rule f(x) = $1 over x$ , defined for real x ≠ 0. Like: Share one whole among the input number of equal portions.
Reciprocal squared function rih-SIP-ruh-kul skwaird FUNK-shun: The rule f(x) = $1 over x squared$ , with real x ≠ 0. All its outputs are positive. Like: Square the divisor before sharing one whole.
Square root skwair root: The nonnegative number whose square equals the given nonnegative input. Like: The side length of a square with a known area.
Principal square root PRIN-sih-pul skwair root: The nonnegative root chosen by the symbol √. Solving a squared equation can require both signs. Like: The nonnegative side length selected from two algebraic signs.
Square root function skwair root FUNK-shun: The rule f(x) = $\sqrt{x}$ . Its real inputs and outputs are both nonnegative. Like: Get a square's side from its area.
Cube root kyoob root: The unique real number whose cube equals the input. Like: Undo a cube-volume calculation.
Cube root function kyoob root FUNK-shun: The rule f(x) = $\sqrt[3]{x}$ . It accepts every real input and can produce every real output. Like: Undo cubing for positive and negative numbers.
Distributive property dih-STRIB-yuh-tiv PROP-er-tee: Multiplying a sum multiplies every term in it: a(b + c) = ab + ac. Like: Several copies of a bag repeat all its contents.
Factor FAK-ter: A quantity multiplied by another quantity in a product. Like: One size of an equal-group package.
Factoring FAK-ter-ing: Rewriting an expression as a product without changing its value. Like: Packing loose pieces into equal groups.
Like terms like termz: Terms with the same variable part and powers, so their coefficients can be combined. Like: Counting apples with apples rather than apples with oranges.
Zero product rule ZEE-roh PROD-ukt rool: If a product equals zero, at least one factor equals zero. Like: One empty factor makes the whole product empty.
Difference quotient DIF-er-ens KWOH-shunt: The average rate of change: new output minus old output, divided by the nonzero input step h. It is the slope between two graph points. Like: Average speed: extra miles divided by extra hours.
Numerator NOO-muh-ray-ter: The top of a fraction, showing the quantity being divided. Like: The pieces you have.
Denominator dih-NOM-uh-nay-ter: The bottom of a fraction, showing the divisor or number of equal pieces in one whole. It cannot be zero. Like: How many equal pieces make one whole.
Power POW-er: A quantity written with a base and exponent. Positive whole-number exponents count multiplied copies; zero and negative exponents extend that pattern. Like: Count how many identical factors are multiplied.
Interval notation IN-ter-vul noh-TAY-shun: A way to write stretches of the number line. Brackets include finite endpoints; parentheses exclude them or accompany infinity. Like: Mark the accepted stretch of a sidewalk.
Union YOON-yun: Combining sets or intervals by accepting values in either one, written ∪. Like: A guest list formed from either of two lists.
Infinity in-FIN-ih-tee: Unending extent, written ∞. It is not a final real-number endpoint. Like: A road with no last mile marker.
Undefined un-dih-FYND: No value is assigned by the rule, often because it would require division by zero. Like: A machine input with no valid instruction.
Percent grade per-SENT grayd: A course score measured out of 100. A grading rule may assign several scores the same letter or grade-point output. Like: A score compared with a full hundred points.
Grade point average grayd point AV-er-ij: A numerical grade measure. Here a grading table assigns grade-point numbers to score bands or letter grades. Like: A smaller set of bins for a larger set of scores.
Radius RAY-dee-us: The distance from the center of a circle to its edge. Here a circle has positive radius. Like: The length of a spoke from a wheel's hub to rim.
Area AIR-ee-uh: The amount of flat surface inside a shape, measured in square units. Like: How many square tiles cover the floor.
π pie: The fixed ratio of a circle's circumference to its diameter, approximately 3.14159. Keep π written for exact circle-area answers. Like: A constant shared by every circle.
Odd od: An integer that is not divisible by 2 into an integer result. Like: Paired chairs with one chair left over.
Even EE-vun: An integer divisible by 2 into an integer result. Like: Chairs that all fit into pairs.
Coefficient koh-uh-FISH-unt: A number multiplying a variable term. Like: How many copies of a quantity you have.
Inequality in-ee-KWAH-lih-tee: A comparison saying one quantity is smaller, larger, or possibly equal. Like: Comparing locations along a number line.
Quadratic expression kwah-DRAT-ik ek-SPRESH-un: A polynomial expression in one variable whose highest power is 2. Like: A recipe containing a squared input as its highest power.
Constant term KON-stunt term: A term without a variable, which keeps its value as the input changes. Like: A fixed fee inside a changing total.
Linear function LIN-ee-er FUNK-shun: A rule multiplying its input by a fixed number, then adding a fixed number: f(x) = mx + b. Like: A fixed fee plus a fixed charge per item.
Origin OR-ih-jin: The point (0, 0), where the horizontal and vertical axes meet. Like: The starting doorway on a map.
Parabola puh-RAB-uh-luh: The graph shape of a quadratic function, appearing as a rounded U or an upside-down U when y depends quadratically on x. Like: A rounded bowl.
Discrete graph dih-SKREET graf: A graph made of separate, isolated points representing its accepted inputs. Like: Individual addresses without a connecting road.
Continuous curve kun-TIN-yoo-us kerv: A graph drawn without gaps along an interval of inputs. Like: An unbroken road through connected locations.
Intersection in-ter-SEK-shun: A point where drawn graphs or lines meet. Touching at one included point counts as one intersection. Like: Two roads meeting at one crossing.
Asymptote AS-im-toht: A line a graph approaches as an input nears an excluded value or extends without bound. The reciprocal toolkit graph approaches both coordinate axes. Like: A boundary a path gets closer to.
Nonnegative non-NEG-uh-tiv: Zero or positive; written x ≥ 0. Like: A location at the doorway or to its right.
Circumference ser-KUM-fer-ens: The distance all the way around a circle's edge. Like: The length of a string wrapped around a wheel.
Diameter dye-AM-uh-ter: The distance across a circle through its center, equal to twice its radius. Like: Two wheel spokes placed end to end through the hub.
Sector SEK-ter: A wedge of a circle enclosed by two radii and the arc between them. Like: One slice of a round pizza.
Circle SER-kul: The set of points at one fixed positive distance from a center. Like: A round hoop around its center.
Exponent EK-spoh-nunt: The raised number telling how many copies of a base are multiplied for a positive integer power. Like: A count of identical factors.
Plane playn: A flat surface extending without end in every direction. Graph points and their two coordinate axes lie in a plane. Like: A sheet of paper imagined to continue beyond every edge.
± plus or MY-nus: A symbol packing both a positive and a negative choice into one line. The choices differ unless the quantity is zero. Like: Two signed choices in one label.
≠ is not EE-kwul too: A symbol saying two quantities are not equal. Like: A label rejecting one forbidden value.
→ goz too: An arrow showing which output is paired with an input. Read it goes to. Like: A route from a chosen button to its result.
Slope slohp: A line's steepness, measured as change in height divided by change in horizontal position. A downward change can give a negative slope. Like: How much a road climbs for each forward step.
Average rate of change AV-er-ij rayt uv chaynj: The average output change for each one unit of input change between two inputs. It equals the slope of the line joining their graph points. Like: Extra miles divided by extra hours gives average speed.
Difference DIF-er-ens: The result of a subtraction. For a change, subtract the old value from the new value. Like: Compare two odometer readings by subtraction.
Quotient KWOH-shunt: The result of a division. The divisor must be nonzero. Like: Share a total among a stated number of groups.
Nonleap year non LEEP yeer: A calendar year with 365 days and 28 days in February. Fixing this kind of year makes each month have one day count. Like: Choose one calendar before counting its month lengths.
Leap year LEEP yeer: A calendar year with 366 days, including February 29. Fix the year before asking for a month's day count. Like: A calendar with one extra day in February.
Polynomial pol-ee-NOH-mee-ul: A sum of constant multiples of whole-number powers of a variable. Its variable is not inside a root or denominator. Like: Several power recipes added together.
Term turm: One added or subtracted piece of an expression. Keep its sign with that piece. Like: One item in a total.
Component kum-POH-nunt: One entry in an ordered pair, also called a coordinate. The first component is input, and the second is output. Like: One part of a two-part address.

Quick checks

For r(x) = 3x − 7, find r(−2).

−13 because 3(−2) − 7 = −6 − 7 = −13.

For k(t) = , solve k(t) = 3.

t = 15 because squaring gives t − 6 = 9, and = 3.

Use the pictured table to solve g(n) = 6.

n = 2.
n = 4.
Both highlighted output cells are 6, so both inputs solve it.

Use the pictured table. Is it a function? Is it one-to-one?

Function: yes, because every input has one output.
One-to-one: no, because inputs 2 and 6 share output 11.

For f(x) = 11 − 6x, simplify , h ≠ 0.

−6, with h ≠ 0, because the output difference is (11 − 6x − 6h) − (11 − 6x) = −6h.

In words, what does measure?

Average output change for each input step: the slope of the line joining the points at x and x + h, because the fraction divides output change by input change.

Read the pictured square-root graph. What are its domain and range?

Domain: [0, ∞).
Range: [0, ∞).
The graph starts at (0, 0), runs right forever, and rises without a largest height.

Does y = + 1 pass the vertical and horizontal line tests on all real inputs?

Vertical: yes, because each input gives one output $x^{2}$ + 1.
Horizontal: no, because inputs −1 and 1 both give output 2.

Before you start

Signed numbers and the order of operations
A number line is a sidewalk with zero at your doorway. Positive numbers take you right. Negative numbers take you left. Adding a negative is a leftward walk. Multiplication counts equal moves, and multiplying by a negative reverses their direction. A power counts repeated multiplication: $3 squared$ means 3 × 3. The raised 2 is the exponent, counting how many copies are multiplied. Parentheses tell you which whole quantity a power or multiplication uses. The order of operations makes everyone read a calculation the same way. A variable is a letter holding a value. Adjacent letters mean multiplication: ah = a × h, and 2ah = 2 × a × h.
Read the math symbols before using them
Math symbols are short road signs. You read the sign before deciding where to go. The equal sign = says two expressions have the same value. The sign ≠ says they do not have the same value. The arrow → means goes to, connecting an input to its output. The sign ± means plus or minus and abbreviates two possibilities. An expression is a number, letter, or calculation such as x + 2. A formula is a rule written with expressions. An equation says two expressions are equal. A term is one piece separated by addition or subtraction; in 3x + 2, the terms are 3x and 2. An input is the value you start with; an output is the result. A function is a rule that assigns one output to each allowed input. A positive integer is a whole counting number such as 1 or 2. A polynomial is a sum of constant multiples of whole-number powers of a variable, such as $x^{2}$ + 3x − 4. Its powers do not have variables in denominators or roots. A constant is a fixed number. A coefficient is a number multiplying a variable term; in 3x, the coefficient is 3.
Fractions, sharing, and safe cancellation
A fraction means division. $6 over 8$ is six pieces when a whole has eight equal pieces. Combining every two eighths gives three quarters, so $6 over 8$ = $3 over 4$ . The denominator is the bottom number and the numerator is the top. To add or subtract, the pieces need a common size. To multiply, multiply the tops and the bottoms. To divide by a nonzero fraction, multiply by its reciprocal, which is the fraction turned over. A factor is a quantity being multiplied.
Distribute, subtract parentheses, and collect like terms
Think of a bag containing two kinds of objects. Three copies of the bag give you three copies of everything inside. That is the distributive property: 3(x + 2) = 3x + 6. A minus sign before a bag takes away all its contents. You must therefore subtract every term inside. Like terms are quantities with the same letter part and power. You can count them together, as you can combine three apples and two apples.
Square a sum by multiplying two copies
Squaring a sum means multiplying the whole sum by itself. Imagine a square floor whose side has two pieces, a and h. Cut the floor at the join in each direction. You get four rectangles, not two: an a-by-a square, two a-by-h rectangles, and an h-by-h square. Their areas add to the whole floor. This picture explains the middle term people often lose.
Equations and inequalities: keep both sides balanced
An equation says two amounts are equal, like the two pans of a balanced scale. A variable is a letter holding an unknown number. To find it, undo operations while doing the same thing to both sides. An inequality compares sizes instead: x > 2 means x is bigger than 2, while x ≥ 2 includes 2. The symbols < and ≤ mean smaller than, and smaller than or equal to. A linear rule multiplies its input by a fixed number and then adds a fixed number, as in 3x − 7.
Pull out a common factor
Imagine several bags that all contain the same number of groups. You can write the shared group count once outside parentheses. Factoring out a common factor reverses distribution. First identify a multiplier in every term. Then divide each term by it to find what belongs inside. The outside multiplier times the new inside must reproduce the entire old sum. This is the step that lets you simplify a difference quotient after the subtraction.
Solve for one letter while another stays
Suppose a bill lists two quantities, but you want a recipe for only one of them. Keep the other letter as a known amount, like the fixed price already written on the bill. To solve for y means get y alone. If you are asked for y as a function of x, x is the input and y is the output. The right side may keep x, but it must not keep y. Equal changes on both sides preserve the relationship.
Factoring and the zero product rule
Factoring is packing a loose sum back into multiplied groups, like returning items to equal bags. Start with numbers: (x + 3)(x − 1) = $x^{2}$ − x + 3x − 3 = $x^{2}$ + 2x − 3. The numbers 3 and −1 add to the middle number 2 and multiply to the last number −3. Factoring runs this backward. A quadratic expression has highest power 2. A coefficient is the number multiplying a term; 2 is the coefficient of x here. The constant term is the number without a letter, which is −3 in this expression. The zero product rule says at least one multiplied factor must be zero when the product is zero.
Squares, roots, and absolute value
A square root undoes a square. Nonnegative means zero or positive. $square root of 9$ means the nonnegative number whose square is 9, so it means 3. This chosen answer is the principal square root. Solving $x^{2}$ = 9 is different: both 3 and −3 work. A cube root undoes three repeated factors. ∛(−8) = −2 because (−2 $) cubed$ = −8. Real numbers are the numbers on the number line. No real number squared is negative. The bars |x| are read the absolute value of x. They ask for the distance from x to zero. Distance records size without direction, so |−3| and |3| both equal 3.
See absolute value as distance
Picture standing on a road marked with zero at your door. A location of −7 is seven steps left. A location of 7 is seven steps right. Both locations are seven steps away from zero. The bars in |x| ask for this distance, not the direction. You keep the size and report it without a minus sign. Zero has distance zero. This is why absolute value is nonnegative.
Sets, coordinates, and interval notation
A set is a collection whose members are called elements. Braces list the members, and repeated entries do not add new members. Natural numbers are 1, 2, 3 and so on; integers also include zero and negatives. An ordered pair (x, y) gives a graph address: x across, then y up or down. Interval notation describes a stretch of the number line. Brackets include an endpoint, and parentheses exclude it. The plane is the flat graph surface. Its horizontal axis is the left-to-right line, and its vertical axis is the up-and-down line. They cross at the origin (0, 0). A coordinate is one number in a point’s address. A coordinate is also called a component of the ordered pair.
Circle radius and area
Picture a round pizza. Its circle is the outline of points all the same distance from the center. That distance is the radius r. The area A is how much flat surface is inside, measured in square units. The distance around the edge is the circumference; the distance straight across the center is the diameter, or 2r. Every circle has the same ratio of circumference to diameter, called π, read pie. It is about 3.14159. Keeping π written preserves exactness. A circle here uses r > 0, so a radius cannot be a negative number.