Relations and Functions

In this chapter: ordered pairs and Cartesian product, relations with domain, codomain and range, functions and real functions, identity, constant, polynomial, rational, modulus, signum and greatest integer functions, and the algebra of real functions.

Cartesian product of sets

An ordered pair (a, b) is a pair in which the order matters, so (2, 3) and (3, 2) are different. Two ordered pairs are equal only when their corresponding entries are equal:

For non-empty sets A and B, the Cartesian product is the set of all ordered pairs with first entry from A and second from B:

• If A or B is empty, A × B = φ.
• A × B ≠ B × A in general, though both have the same number of elements.
• If either A or B is infinite (and the other is non-empty), A × B is infinite.
• A × A × A = {(a, b, c) : a, b, c ∈ A}, a set of ordered triplets. R × R is the coordinate plane and R × R × R is three-dimensional space.

Relations

A relation R from A to B is any subset of A × B. Usually it is described by a rule linking the first element to the second. If (a, b) ∈ R, we say b is the image of a and a is a pre-image of b.

• Domain of R: the set of all first elements of the pairs in R.
• Range of R: the set of all second elements.
• Codomain: the whole set B. Range ⊂ codomain, and the two need not be equal.

If n(A) = p and n(B) = q, then n(A × B) = pq, and every subset of A × B is a relation, so the number of relations from A to B is 2^pq. This count includes the empty relation and A × B itself.

Worked example: A = {1, 2} and B = {3, 4, 5}. Write A × B and find the number of relations from A to B.
Solution: A × B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}, so n(A × B) = 2 × 3 = 6. The number of relations is 2⁶ = 64.

Functions

A relation f from A to B is a function if every element of A has one and only one image in B. Two things can break this: an element of A with no image, or an element of A with two or more images. Elements of B may be left without a pre-image, and two elements of A may share an image; neither stops f from being a function.

We write f : A → B and y = f(x). A is the domain, B the codomain, and the set of images {f(x) : x ∈ A} is the range.

A function whose domain and codomain are subsets of R is a real function. When only a formula is given, take the domain to be the largest set of real x for which the formula gives a real value. In practice: a denominator cannot be zero, and an expression under a square root must be ≥ 0 (strictly > 0 if the root is also in a denominator).

A side result worth knowing: if n(A) = p and n(B) = q, the number of functions from A to B is q^p, since each of the p elements has q choices of image.

Some standard functions and their graphs

The graph of the identity function is the line y = x through the origin at 45°; the constant function is a horizontal line; f(x) = x² is a parabola opening upwards with vertex at the origin, and f(x) = x³ rises through the origin. The reciprocal function f(x) = 1/x, x ≠ 0, has two branches, in the first and third quadrants.

For the greatest integer function, [2.7] = 2, [5] = 5, [−0.4] = −1 and [−2.5] = −3. On a number line, [x] is the integer at or immediately to the left of x.

Algebra of real functions

Let f and g be real functions defined on a common domain X ⊂ R, and let α be a real number. Then, for x ∈ X:

If f and g come with different domains, the new function lives on the intersection of the two domains, and for f/g you also remove the points where g vanishes.

Worked example: f(x) = x² and g(x) = 2x + 1 on R. Find (f + g)(x), (fg)(x) and (f/g)(x).
Solution: (f + g)(x) = x² + 2x + 1 = (x + 1)². (fg)(x) = x²(2x + 1) = 2x³ + x². (f/g)(x) = x²/(2x + 1), defined for x ≠ −1/2.

Finding domain and range

For the domain, list the restrictions and solve them together. For the range, a reliable method is to put y = f(x), solve for x in terms of y, and ask which y give a real x that lies in the domain.

Worked example: Find the domain and range of f(x) = (x + 1)/(x − 2).
Solution: The denominator vanishes at x = 2, so the domain is R − {2}. Put y = (x + 1)/(x − 2). Then xy − 2y = x + 1, so x(y − 1) = 2y + 1 and x = (2y + 1)/(y − 1). This is real for every y ≠ 1. Check that it never gives x = 2: (2y + 1)/(y − 1) = 2 would need 2y + 1 = 2y − 2, which is impossible. So the range is R − {1}.

Two more you should know without working: f(x) = √9 − x² has domain [−3, 3] and range [0, 3]; f(x) = 1/x has domain and range both R − {0}.

Common mistakes: (1) Writing [−2.5] = −2. The greatest integer not exceeding −2.5 is −3. (2) Treating (a, b) and (b, a) as the same pair, or A × B as equal to B × A. (3) Mixing up the counts: relations from A to B number 2^pq, functions number q^p. (4) Calling the codomain the range. (5) Forgetting that for √(x − 5) in a denominator the condition is x > 5, not x ≥ 5. (6) Leaving out the zeros of g in the domain of f/g.

Exam focus

• Equality of ordered pairs and the number of elements in A × B.
• Counting relations and functions between finite sets.
• Deciding from pairs, arrow diagrams or graphs whether a relation is a function.
• Domain and range of rational and square-root expressions, written in interval notation.
• Graphs and values of |x|, sgn x and [x]; these three reappear in limits, continuity and area questions in Class 12.

Practice questions