Relations & Functions | ISC Class 11 Maths Notes

This note covers ordered pairs, Cartesian products, relations, functions, domain, codomain and range, standard real functions and their graphs, and the sum, difference, product and quotient of functions.

What are ordered pairs, and when are they equal?

A set is a well-defined collection of objects; its objects are called elements. An ordered pair consists of two elements written in a specified order inside round brackets. In (a, b), the letters a and b denote the first and second elements respectively.

The order is part of the information. Listing a set in a different order leaves it unchanged, but reversing an ordered pair can change the pair. Compare corresponding positions when pairs are equal.

Property: Equality of ordered pairs

For elements a, b, c and d, (a, b) = (c, d) if and only if a = c and b = d. The phrase if and only if means that the statement works in both directions: pair equality gives component equality, and component equality gives pair equality.

Worked example 1. Find the unknown numbers x and y if (x + 1, y − 2) = (3, 1).

Answer: Equate the first elements to obtain x + 1 = 3, so x = 2. Equate the second elements to obtain y − 2 = 1, so y = 3. Substitution gives (3, 1) on both sides.

The symbol ∈ means “is an element of”. If A and B name two sets, a ∈ A means that a belongs to A, while b ∈ B means that b belongs to B. This notation records where each component of a pair is allowed to come from.

Curly brackets enclose a set; round brackets enclose an ordered pair. Keeping these roles separate helps when a set itself consists of ordered pairs. The pair is one element of that set, even though it has two components.

How are Cartesian products formed and counted?

Definition: The Cartesian product, or cross product, A × B is the set of all ordered pairs whose first element belongs to A and whose second element belongs to B. The symbol × here denotes a Cartesian product of sets.

In symbols, A × B = {(a, b) : a ∈ A, b ∈ B}. The colon means “such that”. To list the product, keep one element of A fixed while pairing it with every element of B, then repeat for each remaining element of A.

Property: Cardinal number of a Cartesian product

A finite set has a definite number of elements, called its cardinal number; otherwise it is infinite. Write n(A) = p and n(B) = q, where n denotes the element count and p and q are those counts. Then n(A × B) = pq, where pq means p multiplied by q.

The empty set, written ∅, contains no elements. If either factor is empty, the Cartesian product is empty. If both factors are non-empty and at least one is infinite, their Cartesian product is infinite.

Worked example 2. For P = {a, b, c} and Q = {r}, where the letters denote distinct elements, form P × Q and Q × P.

Answer: P × Q = {(a, r), (b, r), (c, r)} and Q × P = {(r, a), (r, b), (r, c)}. Each product contains 3 pairs, but the products are unequal because the corresponding pairs have their components reversed.

What do products of real numbers represent?

Let ℝ denote the set of real numbers. The product ℝ × ℝ contains all pairs (x, y) of real numbers and represents all points in a two-dimensional coordinate plane. The coordinates x and y specify positions along the horizontal and vertical axes.

The product ℝ × ℝ × ℝ contains all ordered triplets (x, y, z), where z is the third real coordinate. It represents all points in three-dimensional space. An ordered triplet has three specified positions, just as an ordered pair has two.

Worked example 3. If P = {1, 2}, list P × P × P.

Answer: P × P × P = {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}.

What is a relation, and how are its three sets identified?

A subset of a set is a set whose elements all belong to that set. A relation R from a non-empty set A to a non-empty set B is a subset of A × B. Here R names the relation, while ℝ continues to mean the real numbers.

A relationship between the components selects the pairs that belong to R. In a pair (x, y) belonging to R, y is called an image of x. A relation on A means a relation from A to itself.

Property: The range is a subset of the codomain

Every second component of a pair in R comes from B. Therefore the range is a subset of the codomain. It can equal the codomain, but equality is not required. Similarly, a relation from A need not use every element of A as a first component.

Worked example 4. Let A = {1, 2, 3, 4, 5, 6}. Define R from A to A by y = x + 1, with both x and y in A. Find its pairs, domain, codomain and range.

Answer: R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}. Its domain is {1, 2, 3, 4, 5}, its codomain is {1, 2, 3, 4, 5, 6}, and its range is {2, 3, 4, 5, 6}.

Roster form lists the pairs individually. Set-builder form states the condition that selects them, as in R = {(x, y) : x ∈ A, y ∈ A, y = x + 1} for the preceding example.

Two ovals each contain 1 to 6. Arrows run from 1 to 2, 2 to 3, 3 to 4, 4 to 5 and 5 to 6. The left-hand 6 has no outgoing arrow.

See Fig. 2.5 in your NCERT textbook

An arrow diagram depicts each selected pair by an arrow from its first element to its second. Read the direction before writing the pair. An unused target element still belongs to the codomain, even though no arrow reaches it.

Worked example 5. How many relations are possible from A = {1, 2} to B = {3, 4}?

Answer: A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} has 4 elements. Each relation is a subset of this product, so the number of relations is 2⁴ = 16. Here 2⁴ means four factors of 2 multiplied together. Each pair can be included or excluded independently.

When does a relation qualify as a function?

Definition: A function f from A to B is a relation in which every element of A has one and only one image in B. A function is also called a map or mapping.

Write f: A → B, where the arrow means “maps from A to B”. Here A is the domain and B is the codomain. The notation y = f(x) means that y is the value assigned to input x by the function f.

If f(x) = y, then y is the image of x and x is a preimage of y. The range consists of the images actually obtained. In a function from A to B, the domain is the whole of A, not merely a selected part of it.

Which two checks are necessary?

1. Check that every element of the specified source set A has an image.
2. Check that no element of A has two different images.
3. Allow different inputs to have the same image; this does not violate the definition.
4. Keep the codomain B distinct from the range of images actually reached.

Worked example 6. Consider R = {(2, 1), (3, 1), (4, 2)} on the domain of first components {2, 3, 4}. Does it define a function?

Answer: Yes. The inputs 2, 3 and 4 each have exactly one image. The range is {1, 2}. The fact that both 2 and 3 map to 1 does not prevent the relation from being a function.

By contrast, {(2, 2), (2, 4), (3, 3), (4, 4)} is not a function on its first components because 2 has two different images. The relation y = x + 1 on {1, 2, 3, 4, 5, 6} also fails as a function from that set to itself: 6 has no permitted image.

Let ℕ denote the natural numbers 1, 2, 3 and so on. The relation y = 2x from ℕ to ℕ is a function: each natural number has exactly one double. Its domain and codomain are ℕ, while its range is the set of even natural numbers.

How do domain, range and function values work for real functions?

A real-valued function has real numbers as its values: its range is ℝ or a subset of ℝ. A real function also has ℝ or a subset of ℝ as its domain. The domain tells us which inputs are permitted; the rule tells us their outputs.

When a domain is explicitly given, use that domain. When asked for the domain of a real function given by a formula without a stated restriction, find the real inputs for which the expression is defined. A zero denominator is not permitted.

How is a value calculated?

To calculate f(x) at a specified input, substitute that input wherever x occurs in the rule. The brackets in f(x) indicate the input of the function; they do not mean that a number f is being multiplied by x.

Worked example 7. Let f: ℕ → ℕ be defined by f(x) = 2x + 1. Calculate its values at the natural numbers 1 to 7.

Answer: Substitute each input in 2x + 1. For example, f(1) = 2 × 1 + 1 = 3 and f(7) = 2 × 7 + 1 = 15. Here × between numbers means ordinary multiplication.

This table shows values at seven inputs. It does not change the stated domain ℕ into a finite set. A value table samples the function, whereas the domain and rule specify all its allowed input-output pairs.

Worked example 8. Find the domain of the real function f(x) = (x² + 3x + 5)/(x² − 5x + 4), where x² means x multiplied by itself.

Answer: The denominator factors as (x − 4)(x − 1), which is zero at x = 4 or x = 1. Exclude these two inputs. The domain is ℝ ∖ {1, 4}, where ∖ means set difference: remove the listed elements from ℝ.

The range requires a separate check of attainable outputs. Finding which inputs are excluded does not by itself settle which outputs occur. Always distinguish a restriction on x from a restriction on the function value y.

What do identity and constant functions look like?

The identity function on ℝ is f(x) = x. It returns each input unchanged. Its domain, codomain and range are all ℝ. For every proposed real output, the same real number can serve as an input producing that output.

The graph of a real function is the collection of points (x, f(x)) in the coordinate plane. The horizontal axis is the x-axis and the vertical axis is the y-axis. Their intersection, (0, 0), is called the origin.

The graph labelled f(x) = x is a straight line passing through the origin. It extends from the lower left to the upper right.

See Fig. 2.8 in your NCERT textbook

How does a constant function differ?

A constant is a fixed value. For the constant function f: ℝ → ℝ defined by f(x) = c, the symbol c denotes one fixed real number. Every real input has that same image. The domain is ℝ and the range is the single-element set {c}.

The graph lies at height c and is horizontal. The example f(x) = 3 gives the line y = 3. Every point of this graph has second coordinate 3, even though its first coordinate can be any real number.

The graph labelled f(x) = 3 is a horizontal line through the point marked 3 on the vertical axis. It extends on both sides of the vertical axis.

See Fig. 2.9 in your NCERT textbook

Both are functions because each allowed input receives exactly one output. A range containing just one element is fully compatible with the definition of a function. It is the number of outputs for each individual input that matters.

How are polynomial and rational functions recognised and sketched?

A polynomial function has the form f(x) = a₀ + a₁x + a₂x² + … + aₙxⁿ. Here n is a non-negative integer (one of 0, 1, 2, …); a₀, a₁, …, aₙ are fixed real coefficients, meaning the numbers multiplying the powers of x. The dots continue the pattern of successive whole-number powers.

The symbol xⁿ means x raised to the power n; x² and x³ mean its square and cube. Polynomial functions have domain ℝ. Their ranges depend on the rule. For f(x) = x³, both domain and range are ℝ: every real number has a real cube root.

In interval notation, a square bracket includes an endpoint and a round bracket excludes it. The symbol ∞ means an unbounded positive direction, not a real endpoint; −∞ means an unbounded negative direction. Thus (0, ∞) excludes zero, whereas [0, ∞) includes it.

Worked example 9. For f: ℝ → ℝ defined by f(x) = x², give the values from x = −4 to x = 4 shown below, and identify the domain and range.

Answer: Square each input. The domain is ℝ. The range is [0, ∞), meaning all real numbers greater than or equal to zero. A square cannot be negative, and every non-negative real number is the square of a real number.

The square graph is a U-shaped curve meeting the axes at the origin, with matching left and right branches. The cube graph passes through the origin from the lower left to the upper right.

See Figs. 2.10 and 2.11 in your NCERT textbook

What restriction does a rational function require?

A rational function is a quotient of polynomial functions. In f(x)/g(x), the letters f and g name the numerator and denominator functions. The rule is defined only where g(x) ≠ 0; the symbol ≠ means “is not equal to”.

Worked example 10. Find the domain and range of the reciprocal function f(x) = 1/x.

Answer: The domain is ℝ ∖ {0} because division by zero is undefined. The range is also ℝ ∖ {0}. A reciprocal cannot equal zero, and each non-zero real output y is obtained by choosing x = 1/y.

Two separate curved branches lie above and to the right of the origin, and below and to its left. Each branch draws close to the coordinate axes without crossing them.

See Fig. 2.12 in your NCERT textbook

How do modulus and signum functions treat negative inputs?

The modulus, or absolute value, of a real number x is written |x|. The modulus function is f(x) = |x|. For a non-negative input, its value equals the input; for a negative input, its value is the negative of that input.

In symbols, |x| = x when x ≥ 0, and |x| = −x when x < 0. The symbols ≥ and < mean “greater than or equal to” and “less than”. In the negative-input case, −x is positive: the minus sign reverses the sign of the negative input.

The domain of the modulus function is ℝ and its range is [0, ∞). Zero is included because |0| = 0. Its graph meets the origin and consists of two straight arms, one on each side of the vertical axis.

The graph is V-shaped with its meeting point at the origin. Its left arm rises towards the upper left, and its right arm rises towards the upper right.

See Fig. 2.13 in your NCERT textbook

What information does the signum function retain?

The signum function reports whether the input is negative, zero or positive. It assigns −1 to a negative input, 0 to zero and 1 to a positive input. It therefore records the sign of a number rather than its magnitude.

The signum function has domain ℝ and range {−1, 0, 1}. Although many inputs receive the same value, each individual input has just one value. It therefore satisfies the function definition.

A horizontal ray at y = 1 extends to the right with an open endpoint at the vertical axis. Another at y = −1 extends left with an open endpoint. The value at x = 0 is separately specified as 0.

See Fig. 2.14 in your NCERT textbook

A rule with different formulas for different input conditions is a piecewise-defined function. Read the condition before using its formula. The three signum conditions separate negative, zero and positive inputs without assigning two different values to a single input.

How does the greatest integer function produce a step graph?

The greatest integer function is f(x) = [x], where [x] means the greatest integer less than or equal to the real number x. Integers are the numbers …, −2, −1, 0, 1, 2, …; their set is denoted by ℤ.

The brackets in [x] denote this function, not an interval. Its output is the greatest integer at or below the input. The symbol ≤ means “less than or equal to”.

Each interval includes its left endpoint and excludes its right endpoint. On reaching the next integer, the function takes that integer as its value and begins the next horizontal step.

How are the domain and range read?

Every real number has a greatest integer at or below it, so the domain is ℝ. The outputs are integers, and each integer occurs as an output at that integer itself. Therefore the range is ℤ.

The graph consists of horizontal steps at integer heights. The displayed steps have open circles at their right endpoints. Each step extends across one unit along the horizontal axis.

See Fig. 2.15 in your NCERT textbook

A filled endpoint means that the point belongs to the graph; an open endpoint means that it is excluded. This distinction keeps the two neighbouring steps from assigning two values at their shared boundary input.

For a negative input between −1 and 0, the answer is −1, not 0. Zero is greater than that input, so it fails the requirement “less than or equal to”. Keeping this direction of the inequality is essential when reading the negative steps.

What are exponential and logarithmic functions?

In an exponential function, the input appears in the exponent: f(x) = bˣ, where b is a fixed positive base. The base is the number being raised to the power x. First consider b > 1.

The domain is ℝ and the range is (0, ∞). The graph passes through (0, 1), since b⁰ = 1, and is increasing: its height rises as the input moves from left to right. For very large negative inputs, it approaches the horizontal axis but never meets it.

How is a logarithm related to a power?

The logarithm logᵦ x is the exponent to which b must be raised to obtain x. Thus y = logᵦ x means bʸ = x. The small b identifies the base. For b > 1, the logarithmic function has domain (0, ∞) and range ℝ.

Its graph passes through (1, 0) and is increasing. As a positive input approaches zero, its logarithm becomes smaller than any prescribed real number. The graph approaches the vertical axis without meeting it. Non-positive inputs have no real logarithm.

Worked example 11. Rewrite the power equalities 2³ = 8 and 10⁴ = 10000 as logarithmic equalities.

Answer: 2³ = 8 gives log₂ 8 = 3. Similarly, 10⁴ = 10000 gives log₁₀ 10000 = 4. In each case, the logarithm is the exponent, and the base remains the same.

Base 10 gives the common exponential function and common logarithms. The constant e is a particular real number between 2 and 3 used as the natural base. The functions eˣ and logₑ x are called the natural exponential and natural logarithmic functions. Often natural logarithm is denoted by ln.

The exponential curve passes through (0, 1), and the logarithmic curve passes through (1, 0). They are mirror images in the dotted straight line y = x.

See Fig. 5.11 in your NCERT textbook

What changes when the base is between zero and one?

The standard base conditions are b > 0 and b ≠ 1. When 0 < b < 1, both graphs are decreasing, meaning their height falls as the input moves right. Their domains, ranges and the respective points (0, 1) and (1, 0) remain the same.

Base 1 would give the constant value 1 for every exponential input, so it cannot determine a unique logarithmic exponent. Always state the base condition when describing whether an exponential or logarithmic graph increases or decreases.

How are functions added, subtracted, multiplied and divided?

Let f and g be real functions defined on a common set X of real inputs. Their sum, difference and product are formed by combining their values at the same input x. This is called pointwise calculation.

The notation fg names the product function; f/g names the quotient function, obtained by division. A scalar here is a fixed real number k. Scalar multiplication gives (kf)(x) = kf(x), multiplying every output by the same number.

How are formulas combined without losing restrictions?

Worked example 12. Let f(x) = x² and g(x) = 2x + 1, both defined on ℝ. Find their sum, difference, product and quotient.

Answer: (f + g)(x) = x² + 2x + 1. Also, (f − g)(x) = x² − 2x − 1 and (fg)(x) = x²(2x + 1) = 2x³ + x². These three functions have domain ℝ.

The quotient is (f/g)(x) = x²/(2x + 1). The condition 2x + 1 ≠ 0 excludes x = −1/2, so its domain is ℝ ∖ {−1/2}.

Subtracting g(x) requires subtracting its entire expression, including its constant term. In the quotient, excluding a zero of g is part of defining the new function. It is not an optional condition to add after using the formula.

Worked example 13. Let f(x) = √x and g(x) = x on [0, ∞), where √x denotes the non-negative square root of x. Find the four algebraic combinations.

Answer: The sum is √x + x, the difference is √x − x, and the product is x√x, all for x ≥ 0. The quotient is √x/x = 1/√x, with domain (0, ∞), because g(0) = 0.

Note: If the original functions have different domains, first use the inputs allowed by both. For a quotient, additionally remove inputs at which the denominator function is zero. Simplifying an expression does not restore an input excluded from the original quotient.

A complete answer therefore has two parts: the resulting expression and the inputs on which it defines the requested function. Keep both in view when comparing functions or using their graphs.

Glossary

• Ordered pair — Two elements grouped in a specified order, with distinct first and second positions.
• Cartesian product — The set of all ordered pairs formed by choosing components from two sets in order.
• Cardinal number — The number of elements in a finite set, used to count a finite Cartesian product.
• Relation — A subset of a Cartesian product that associates first components with second components.
• Domain — The set of first components of a relation, or all permitted inputs of a function.
• Codomain — The specified target set within which all images of the relation or function lie.
• Range — The set of actual second components, or outputs, obtained from a relation or function.
• Function — A relation assigning exactly one image to every element of its specified source set.
• Preimage — An input that a function maps to a particular output in its range.
• Real function — A function whose domain and range are each sets of real numbers.
• Modulus function — The function that returns the absolute value of each real input.
• Greatest integer function — The function returning the greatest integer less than or equal to each real input.
• Exponential function — A function with the input as exponent and a fixed positive base different from one.
• Logarithmic function — A function returning the exponent required to obtain its positive input from a specified base.
• Quotient of functions — The function formed by dividing corresponding values, excluding inputs where the denominator function is zero.

Common errors and misconceptions

• Misconception: Reversing the entries of an ordered pair leaves it unchanged. Correct: First and second positions matter. Equality requires equality of the corresponding components, so reversing unequal components changes the pair.
• Misconception: A × B and B × A must be the same because their finite element counts agree. Correct: Equal counts do not imply equal sets of ordered pairs.
• Misconception: The domain of every relation from A to B must be A. Correct: Its domain contains the first elements actually used. A function from A to B must use every element of A.
• Misconception: Range and codomain are interchangeable. Correct: The codomain is specified in advance; the range contains actual images and is a subset of that codomain.
• Misconception: Different inputs cannot share an output in a function. Correct: Shared outputs are allowed. The forbidden situation is a single input assigned two different outputs.
• Misconception: The expression −x in the modulus rule produces a negative answer. Correct: That branch applies when x is negative, so −x is positive.
• Misconception: The greatest integer function rounds to the nearest integer. Correct: It chooses the greatest integer at or below the input, including for negative inputs.
• Misconception: A quotient of functions keeps every input in their common domain. Correct: Inputs making the denominator zero must be excluded, even when later algebra simplifies the expression.

Exam-style questions with model answers

Q1. If (x + 1, y − 2) = (3, 1), find the unknown numbers x and y using equality of ordered pairs. [2 marks]

1. Equating first components gives x + 1 = 3. Subtracting 1 from both sides gives x = 2.
2. Equating second components gives y − 2 = 1. Adding 2 to both sides gives y = 3.

Q2. Let P = {a, b, c} and Q = {r}, where all four letters denote distinct elements. List P × Q and Q × P, and explain whether they are equal. [3 marks]

1. P × Q = {(a, r), (b, r), (c, r)}. Each first component comes from P, while its second component is the sole element r of Q.
2. Q × P = {(r, a), (r, b), (r, c)}. Reversing the order of the sets reverses which set supplies each component.
3. The products are not equal. For example, (a, r) differs from (r, a) because a and r are distinct, although both products contain three elements.

Q3. Let A = {1, 2, 3, 4, 5, 6}. A relation R from A to A contains exactly those pairs (x, y) with y = x + 1. List R, state its domain, codomain and range, and decide whether it is a function from A to A. [5 marks]

1. R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}. These are the pairs satisfying the rule with both components in A.
2. The domain is {1, 2, 3, 4, 5}, obtained by collecting the first components of the listed pairs.
3. The codomain is A = {1, 2, 3, 4, 5, 6}, because the relation is specified as going from A to A.
4. The range is {2, 3, 4, 5, 6}, the set of second components that actually occur in the relation.
5. R is not a function from A to A. The input 6 has no image in A, so not every element of the stated source set is assigned an image.

Q4. The following relations have their first-component sets as their specified domains: R₁ = {(2, 1), (3, 1), (4, 2)} and R₂ = {(2, 2), (2, 4), (3, 3), (4, 4)}. Here R₁ and R₂ are labels for the two relations. Classify each as a function or not, and give the domain and range of R₁. [4 marks]

1. R₁ is a function because every input in its specified domain has exactly one image. Two different inputs sharing image 1 is allowed.
2. The domain of R₁ is {2, 3, 4}, formed from its first components.
3. The range of R₁ is {1, 2}, formed from its second components without repeating the value 1.
4. R₂ is not a function because the same input 2 is assigned the two different outputs 2 and 4.

Q5. For the real function f(x) = (x² + 3x + 5)/(x² − 5x + 4), factor the denominator, identify the excluded real inputs and state the domain. [3 marks]

1. The denominator factors as x² − 5x + 4 = (x − 4)(x − 1). A rational function is defined only where its denominator is non-zero.
2. The factor x − 4 vanishes at x = 4, and x − 1 vanishes at x = 1. Both inputs must be excluded.
3. The domain is ℝ ∖ {1, 4}: all real numbers except 1 and 4. No other real input makes either denominator factor zero.

Q6. Let f and g be defined for all real x by f(x) = x² and g(x) = 2x + 1. Find their sum, difference and product, then give their quotient and its domain. [5 marks]

1. The sum is (f + g)(x) = f(x) + g(x) = x² + 2x + 1, obtained by adding the outputs at the same input.
2. The difference is (f − g)(x) = x² − (2x + 1) = x² − 2x − 1. The minus sign applies to both terms of g(x).
3. The product is (fg)(x) = x²(2x + 1) = 2x³ + x². The sum, difference and product are defined on all real inputs.
4. The quotient is (f/g)(x) = x²/(2x + 1), formed by dividing the output of f by the output of g.
5. For the quotient, 2x + 1 must be non-zero, so x ≠ −1/2. Its domain is therefore ℝ ∖ {−1/2}.

Q7. Define the greatest integer function by f(x) = [x], the greatest integer less than or equal to real x. State its value on −1 ≤ x < 0 and on 0 ≤ x < 1, and give its domain and range. [4 marks]

1. On −1 ≤ x < 0, the value is −1. Zero cannot be selected because it is greater than each input in this interval.
2. On 0 ≤ x < 1, the value is 0. The next integer, 1, is excluded by the input condition.
3. The domain is ℝ because every real input has a greatest integer less than or equal to it.
4. The range is ℤ because the output is an integer, and every integer occurs as the output when that integer is the input.

Q8. For a fixed real base b > 1, compare the domain and range of the exponential function f(x) = bˣ and the logarithmic function g(x) = logᵦ x. State a point on each graph. [3 marks]

1. The exponential function has domain ℝ and range (0, ∞). Every real exponent is allowed, and the resulting value is positive.
2. The logarithmic function has domain (0, ∞) and range ℝ. Its input must be positive, while its output is a real exponent.
3. The exponential graph contains (0, 1), since b⁰ = 1. The logarithmic graph contains (1, 0), since the exponent required to obtain 1 is zero.

Key takeaways

• An ordered pair preserves component order; equal pairs have equal first components and equal second components.
• A Cartesian product includes every permitted pairing, and its finite element count is the product of the two set sizes.
• A relation is a subset of a Cartesian product; its domain and range collect the components actually used.
• A function assigns exactly one image to every input in its specified domain, while allowing different inputs to share outputs.
• The range contains actual outputs and is a subset of the codomain; the two sets need not be equal.
• Standard function graphs must be read with their domains, ranges and endpoint conditions, especially for reciprocal and step functions.
• An exponential function with base greater than one increases; its logarithmic partner accepts positive inputs and returns real exponents.
• Function operations combine values at the same input; a quotient additionally excludes inputs making the denominator function zero.

Test yourself

What does the first component of a pair in A × B tell you?

It is an element of A. The second component must belong to B, so the order of the two factor sets controls the order of the components.

What happens to A × B if B is empty?

The product is empty because no second component can be chosen from B, so no ordered pair can be formed.

Why can a relation from A to B have a domain smaller than A?

A relation uses selected pairs from A × B. Some elements of A need not occur as first components in any selected pair.

Does a constant function fail the definition because all its outputs agree?

No. Each input still has exactly one image. The function definition allows different inputs to have the same output.

What is the range of f(x) = 1/x on the non-zero real numbers?

The range is the non-zero real numbers. Zero is not a reciprocal, and every non-zero output y is obtained at input 1/y.

How do modulus and signum outputs differ for a negative input x?

The modulus output is −x, a positive magnitude. The signum output is −1, which records that the input is negative.

What do filled and open endpoints mean on a greatest integer graph?

A filled endpoint is included in the graph, while an open endpoint is excluded. Each horizontal step includes its left endpoint and excludes its right endpoint.

Why must the quotient f/g exclude a common-domain input where g(x) = 0?

At that input the formula would require division by zero. The input is therefore excluded from the quotient function's domain.