Domain and range

First you separate the inputs a function accepts from the outputs it reaches. You learn to find allowed inputs by checking the arithmetic, then write those inputs with inequalities, sets and intervals. Next you read the two sets from graphs and explain the behavior of all nine toolkit functions. Finally you use different formulas on different input intervals, choose the correct rule at a boundary, and build the resulting graph.

Lessons

Vocabulary

Function: A rule assigning exactly one output to each allowed input. Different inputs may share an output. Like: A vending machine with one assigned snack for each button.
Input: The value you choose or supply to a function before its rule calculates an output. Like: The button you press on a vending machine.
Output: The result a function produces from a chosen allowed input. Like: The snack delivered after you choose a button.
Independent variable: The variable representing the input. You choose its allowed value before calculating the output. Like: The number of tickets you decide to buy.
Dependent variable: The variable representing the output. Its value depends on the chosen input and the function's rule. Like: The total cost following from your ticket count.
Function notation: A function name followed by an input in parentheses, such as f(3). It means the output at that input. Like: A machine name followed by the button you selected.
Equation: A statement that two expressions are equal. Solving it finds values making the statement true. Like: Two pans of a scale holding equal amounts.
Formula: A mathematical recipe describing how quantities are related or how to calculate a result. Like: A cooking recipe turning ingredients into a dish.
Domain: The set of all inputs allowed for a function, considering its arithmetic and any stated context or restrictions. Like: The choices a vending machine accepts.
Range: The set of all outputs a function actually produces from its allowed inputs. Like: The snacks the machine can deliver.
Set: A collection of distinct objects or values. Its members may be listed or described by a condition. Like: A basket holding items that belong together.
Element: One member of a set. Repeated appearances do not create additional members. Like: One item inside a basket.
Ordered pair: Two values written in a fixed order, usually (x, y): input first, output second. Reversing them usually changes the point. Like: An address giving blocks right, then blocks up.
Coordinate: One number describing a point's position along an axis. In (x, y), x is horizontal and y is vertical. Like: One part of a map address.
Real number: A number with a position on the number line, including whole numbers, fractions, decimals and numbers such as $square root of 2$ . Like: Any location along one straight measuring tape.
Discrete function: A function with separate allowed inputs, such as whole-number counts. Its graph has isolated points rather than every intervening input. Like: Counting people instead of measuring flowing water.
Continuous function: A function with no break at its domain points. On an interval, its graph has no jumps or holes between allowed inputs. Like: A smoothly moving thermometer reading as time passes.
Restriction: A condition limiting allowed inputs, from arithmetic, a stated rule, or a meaningful physical situation. Like: A sign saying which visitors may enter.
Undefined: Having no assigned value in the number system or function being used. Division by zero is undefined in real arithmetic. Like: A machine instruction that cannot produce a result.
Numerator: The top expression in a fraction. It gives the amount divided by the denominator. Like: The count of equal pizza slices you have.
Denominator: The bottom expression in a fraction. It is the divisor and cannot equal 0. Like: The number of equal slices making the whole pizza.
Polynomial: A sum of constant multiples of nonnegative whole-number powers of the variable, such as $x^{2}$ − 3x + 1. Real inputs cause no arithmetic restrictions. Like: A recipe using addition and repeated multiplication.
Inequality: A comparison using <, >, ≤ or >=. It describes which amount is smaller or larger and whether equality is allowed. Like: Comparing positions on a measuring tape.
Compound inequality: Two or more comparisons joined with and or or. And requires every condition; or requires at least one. Like: Passing both entry checks or choosing either entrance.
Nonnegative: Greater than or equal to 0. It includes positive numbers and 0, but excludes negative numbers. Like: A bank balance with no debt, including an empty account.
Positive: Strictly greater than 0. Zero is excluded. Like: A thermometer reading above zero.
Negative: Strictly less than 0. Negative numbers lie to the left of 0 on a number line. Like: A thermometer reading below zero.
Radical: A root symbol or an expression using one. The symbol √ requests the principal square root. Like: A label asking you to undo repeated multiplication.
Radicand: The number or expression inside a root symbol. An even root requires a nonnegative real radicand. Like: The contents under the root symbol's roof.
Even root: A root undoing an even power, such as a square or fourth power. Real even roots require nonnegative radicands. Like: Recovering a square tile's side from its area.
Odd root: A root undoing an odd power, such as a cube. It accepts negative radicands and keeps their negative sign. Like: Undoing three repeated factors while keeping their direction.
Principal square root: The nonnegative square root chosen by the √ symbol. It is one value even when a squared equation has two solutions. Like: Choosing a square tile's nonnegative side length.
Number line: A straight line placing real numbers in order, with smaller numbers left and greater numbers right. Like: A measuring tape extending both ways from zero.
Inequality notation: Writing allowed values with comparison signs, optionally joining conditions with and or or. Like: Giving a measuring tape's allowed positions as comparisons.
Set-builder notation: Describing a set with a condition inside braces, such as {x | x > 3}, read as x such that x exceeds 3. Like: Writing a basket's membership rule instead of listing everything inside.
Braces: The symbols { and } used to enclose a set's listed members or membership condition. Like: The sides of a basket around its contents.
Such that: The phrase introducing the condition members satisfy in set-builder notation. A vertical bar can stand for this phrase. Like: Saying who may enter, followed by the entry condition.
Interval notation: Describing a continuous stretch of real numbers with its limits. Brackets include endpoints; parentheses exclude them. Infinity always takes a parenthesis. Like: Marking a measuring tape's allowed stretch by its two ends.
Interval: A set containing every real number between its limits, with endpoints included or excluded as specified. It may extend without bound. Like: One unbroken stretch of a measuring tape.
Endpoint: A finite boundary value at an interval's end, included or excluded according to its bracket or parenthesis. Like: An end marker of a permitted sidewalk stretch.
Lower limit: The smaller boundary of an interval, written first. It may be −∞ when there is no finite lower bound. Like: The left boundary on a measuring tape.
Upper limit: The greater boundary of an interval, written second. It may be ∞ when there is no finite upper bound. Like: The right boundary on a measuring tape.
Inclusive: Including the boundary value. Use ≤ or ≥ in a comparison, a bracket in interval notation, or a filled dot. Like: An entry sign allowing someone exactly at the required height.
Exclusive: Excluding the boundary value. Use < or > in a comparison, a parenthesis in interval notation, or a hollow dot. Like: An entry sign requiring more than the marked height.
Bracket: The interval symbols [ and ], indicating that the adjacent finite endpoint belongs to the set. Like: A closed boundary keeping its end position inside the allowed stretch.
Parenthesis: In interval notation, ( or ) excludes the adjacent finite endpoint. It also appears beside infinity, which is never an included number. Like: A hollow end marker leaving the boundary out.
Infinity: The symbol ∞ describes continuing without a finite bound. It is not a real number or an endpoint you can include. Like: A road continuing beyond every distance marker you name.
Unbounded: Having no finite limit in at least one direction. A set can be bounded on one side and unbounded on the other. Like: A walkway with a start but no last distance marker.
Union (∪): The set of elements belonging to either set or both. Shared elements are listed once, and the sets may overlap. Like: Combining two guest lists and removing duplicate names.
Intersection: The set of elements shared by every specified set. For conditions, it keeps values satisfying all the conditions together. Like: The names appearing on both guest lists.
Closed dot / open dot: A filled dot includes the point. A hollow dot excludes it. Excluded points do not count in the vertical line test. Like: A filled seat is present; an empty outline is left out.
Graph: A picture of points representing input and output pairs, usually with inputs horizontal and outputs vertical. Like: A map showing every input's output address.
x-axis: The horizontal axis measuring x-values, usually function inputs. Every point on it has y = 0. Like: The east-to-west ruler on a map.
y-axis: The vertical axis measuring y-values, usually function outputs. Every point on it has x = 0. Like: The north-to-south ruler on a map.
Vertical line test: A graph represents a function if every vertical line meets at most one included point. Two crossings give one input two outputs. Like: Holding one button fixed and checking it never has two assigned answers.
Toolkit function: One of the basic functions whose formulas, shapes, domains and ranges help you understand more complicated functions. Like: A familiar tool you recognize before tackling a larger job.
Constant function: A function with the same output for every allowed input. The toolkit version f(x) = c accepts all real inputs and has range {c}. Like: A machine dispensing the same item for every button.
Identity function: The function f(x) = x. Its output equals its input, and its domain and range are both all real numbers. Like: A machine handing back what you put in.
Absolute value function: The function f(x) = |x| gives the input's distance from 0. Its domain is all real numbers and its range is [0, ∞). Like: Measuring your distance from a door, regardless of direction.
Magnitude: A real number's size without its sign, measured by its absolute value or distance from 0. Like: How far you walked, without saying east or west.
Modulus: Another name for the absolute value of a real number, its nonnegative distance from 0. Like: A distance label ignoring which direction you traveled.
Quadratic function: A polynomial whose highest power is $x^{2}$ . The toolkit quadratic f(x) = $x^{2}$ has domain all real numbers and range [0, ∞). Like: Finding a square tile's area from its side length.
Cubic function: A polynomial whose highest power is $x^{3}$ . The toolkit cubic f(x) = $x^{3}$ has domain and range both equal to all real numbers. Like: Finding a cube's volume from a side length.
Reciprocal function: The toolkit function f(x) = $1 over x$ . Zero is excluded from both its domain and range. Like: Sharing one whole by a chosen nonzero amount.
Reciprocal squared function: The toolkit function f(x) = $1 over x squared$ . Its domain excludes 0, and its range contains exactly the positive real numbers. Like: Sharing one whole by a squared, therefore positive, nonzero amount.
Square root function: The toolkit function f(x) = $\sqrt{x}$ . It chooses the principal square root, so its domain and range are [0, ∞). Like: Recovering a square tile's side length from its area.
Cube root function: The toolkit function f(x) = $\sqrt[3]{x}$ . It undoes cubing and has domain and range both equal to all real numbers. Like: Recovering a cube's side from its volume, extended to signed numbers.
Odd function: A function with a domain symmetric about 0 and f(−x) = −f(x). Reversing an input's sign reverses its output's sign. Like: Reflecting horizontal and vertical movements through their starting point.
Piecewise function: A function using different formulas on specified parts of its domain. Each allowed input must have exactly one output. Like: A price list with different rules for different purchase amounts.
Piece: One formula together with the input condition telling you where it is used in a piecewise function. Like: One row of a price list and its qualifying amounts.
Boundary: An input where an interval starts or ends, or where a piecewise condition changes. Check which condition includes it. Like: The marked height where an entry rule changes.
Tax bracket: An income interval with its assigned tax rule. In a marginal model, a higher rate applies only to income within its bracket. Like: Different shelves charging different rates only for the amount on each shelf.
Parabola: The bowl-shaped curve that graphs a quadratic function. It may open upward or downward. Like: A bowl or an upside-down bowl.
Substitution: Replacing every occurrence of a variable with one specified value, then following the formula's operations. Like: Making a recipe with the ingredient amount you selected.
Linear equation: An equation whose simplified variable terms have only the first power, such as 3x + 8 = 29. Like: A balance with equal copies of one unknown weight.
Factor: One of the numbers or expressions multiplied together to form a product. Like: One ingredient in a multiplication recipe.
Factoring: Rewriting an expression as a product of factors without changing its value. Like: Finding the pieces that rebuild a tiled floor.
Difference of squares: One squared expression minus another. The pattern $a^{2}$ − $b^{2}$ factors as (a − b)(a + b). Like: Comparing the areas of two square tiles.
Zero product property: A product of real factors is zero exactly when at least one factor equals zero. Like: Zero boxes or zero items per box both give zero items.
Coefficient: The number multiplying a variable or its power. An unwritten coefficient in x is 1. Like: The number of identical copies in a repeated amount.
Constant term: A fixed added or subtracted number in an expression, without a variable attached. Include its sign. Like: The fixed part of a bill before a per-item charge.
Monic quadratic: A quadratic expression with coefficient 1 on $x^{2}$ . Its factor-pair search matches the middle coefficient and constant term. Like: One main square tile before adding the smaller pieces.
Cancellation: Dividing a numerator and denominator by the same nonzero common factor. Original denominator exclusions remain. Like: Shortening a recipe without changing which choices it accepts.
Nonpositive: Less than or equal to zero. It includes negative numbers and zero, and excludes positive numbers. Like: A temperature at freezing or below on a scale whose zero is freezing.
Domain construction: Creating a formula whose arithmetic permits exactly the requested inputs, then verifying that complete set. Like: Building a gate for a specified guest list.

Quick checks

Find the domain of f(x) = .

(−∞, 4) ∪ (4, ∞), because x − 4 = 0 at x = 4 and division by zero is undefined.

Find the domain of f(x) = .

[9, ∞), because x − 9 ≥ 0 gives x ≥ 9 and the endpoint produces .

Write −1 < x ≤ 6 in interval notation.

(−1, 6], because −1 is excluded and 6 is included.

Write x < −3 or x ≥ 2 in interval notation.

(−∞, −3) ∪ [2, ∞), because or combines the two allowed pieces without filling the gap.

Find the range of f(x) = + 5 for all real x.

[5, ∞), because ≥ 0 and every y ≥ 5 is reached by x = .

Find the domain of f(x) = − 7x + 1.

(−∞, ∞), because a polynomial uses only arithmetic defined for every real input.

Solve 12 − 3x ≥ 0.

x ≤ 4, because subtracting 12 gives −3x ≥ −12 and dividing by negative 3 reverses the comparison.

For f(x) = if x ≤ 1, f(x) = 3 if 1 < x ≤ 2, and f(x) = x if x > 2, find f(2). The input is 2; choose its condition and find the output.

f(2) = 3, because 2 belongs to 1 < x ≤ 2, while the identity branch requires x > 2.

Before you start

Signed numbers, order and moving on a number line
Picture a sidewalk with your front door marked 0. A Number line puts numbers in order along that sidewalk. Positive numbers lie right of 0; Negative numbers lie left. Zero belongs to neither group. Adding a positive moves right, and adding a negative moves left. Subtraction undoes addition, so subtracting a negative moves right. Every Real number has a position, including fractions and decimals between whole numbers. A position farther right is greater. The problem −6 + 9 − (−2) asks where you finish after those moves.
Function notation, substitution, parentheses and order of operations
Imagine a vending machine following one recipe for each button. A Function gives exactly one Output for each allowed Input. Function notation names the machine and the chosen input: f(4) means the output when you put 4 into f. It does not mean f × 4. A Formula gives the recipe. Substitution means replacing every occurrence of the input letter with the chosen number. Parentheses keep that number together, especially if it is negative. Work inside parentheses first, then powers, then multiplication and division, then addition and subtraction. At the same level, work from left to right.
Fractions, decimals and percentages
Think of dividing a pizza into equal slices. A fraction's Denominator tells you how many equal slices make the whole. Its Numerator tells you how many slices you have. $3 over 4$ also means 3 ÷ 4. A decimal writes parts using tenths, hundredths and so on. A percentage counts parts out of 100, so 75% = $75 over 100$ = 0.75. To add or subtract fractions, first make the slice sizes match. To multiply, multiply the tops and the bottoms. We will find $1 over 4$ + $1 over 2$ of an $80 budget, then express that portion as a decimal and percentage.
Solving a linear equation
An Equation says two amounts are equal. Think of the two pans of a balanced scale. Solving means finding the number that keeps the scale balanced. A linear equation has the unknown multiplied by a number and combined with addition or subtraction, without squaring or cubing the unknown. Undo the operations in reverse order from the recipe. Add or subtract the same amount on both sides. Then divide both sides by the same nonzero number. For 3x + 8 = 29, the question is which number, tripled and increased by 8, becomes 29.
Solving inequalities from zero, including the reason negative multiplication or division reverses order
An Inequality compares amounts rather than saying they are equal. Picture two positions on the same sidewalk. The point farther right is greater. The signs < and > mean less than and greater than. The signs ≤ and ≥ also allow equality. Adding or subtracting the same amount moves both points together, preserving their order. Multiplying or dividing by a positive number stretches or shrinks their distances without swapping sides. A negative factor reflects them across 0, so their order reverses. Solve 11 − 3x ≥ −4 by undoing operations while keeping the comparison true.
Compound inequalities: and, or, and keeping both restrictions
Imagine a ride requiring you to be at least one height but below another. You must pass both checks. A Compound inequality connects comparisons with and or or. And keeps only numbers satisfying every condition. This shared part is an Intersection. Or accepts numbers satisfying either condition, including both. That combination is a Union (∪). Sometimes you must also skip a particular number. We will keep −1 ≤ x < 6 and x ≠ 2. The first condition allows numbers from −1 through values below 6; the second removes 2 from that allowed collection.
Squares, even roots, principal square roots and odd roots
Picture a square tile. Squaring its side gives its area: $6 squared$ = 6 × 6 = 36. A root asks for the repeated factor making a number. The Radical sign √ asks for a square root; its inside is the Radicand. The Principal square root is the nonnegative answer selected by √. An Even root cannot take a negative real radicand because an even number of negative factors produces a positive result. An Odd root can: three negative factors produce a negative result. We will compare $square root of 36$ , solutions of $x^{2}$ = 36, and ∛(−216).
Factoring a difference of squares and locating zeros
Think of rebuilding a rectangular floor from smaller tiles. Expanding multiplies the pieces together and adds their areas. Each multiplied piece is a factor. Factoring reverses the process: it rewrites a sum or difference as a product. A difference of squares has one squared amount minus another, such as $x^{2}$ − 25 = $x^{2}$ − $5 squared$ . It factors as (x − 5)(x + 5). Locating zeros means finding inputs making the whole expression equal 0. A product is 0 when at least one factor is 0. We will factor $x^{2}$ − 25 and find its zeros before using such expressions as denominators.
Coordinates, plotting a line from two points, and the vertical line test
A Graph maps input and output pairs. An Ordered pair (x, y) gives an address: x moves right or left, then y moves up or down. Each number is a Coordinate. The horizontal x-axis measures x; the vertical y-axis measures y. To draw y = 2x − 3, choose two inputs, calculate their outputs, plot the addresses and draw the straight line through them. The Vertical line test asks whether one input has two outputs. A vertical line holds x fixed, so two crossings mean different y-values for the same input, which cannot define a function.