Exercise practice: Relations and Functions

4.1 How the exercises are laid out

Exercise 2.1 builds Cartesian products and ordered pairs, Exercise 2.2 builds relations from formulas and pictures, Exercise 2.3 tests functions and asks for domains and ranges, and the miscellaneous exercise adds algebra of functions and mixed proofs.

This page takes one or two representative questions from each exercise and writes the full working. The remaining parts of each question follow the same pattern.

4.2 Exercise 2.1: products and ordered pairs

Question 6 prints the full product as a roster of four pairs and asks you to recover the two sets. Collect the first coordinates to get A = {a, b} and the second coordinates to get B = {x, y}. When reading coordinates, remove repeats, since each set lists its elements once.

Question 8 uses A = {1, 2} and B = {3, 4}. The product is {(1, 3), (1, 4), (2, 3), (2, 4)} and has four elements, so it has 16 subsets. Listing all 16 is possible but tedious; a list by size is safer than a random list.

Question 10 says A x A has 9 elements and contains (-1, 0) and (0, 1). Since 9 = 3 squared, A has three elements. The given pairs put -1, 0 and 1 in A, so A = {-1, 0, 1}. The remaining pairs come from pairing each element with all three, giving nine pairs in total.

Question 7 verifies A x (B intersection C) = (A x B) intersection (A x C) for A = {1, 2}, B = {1, 2, 3, 4} and C = {5, 6}. Here B intersection C is empty, so the left side is empty. The right side is the intersection of a product with first entries from B and a product with first entries from C, which share no pair, so it is empty too. The verification goes through the definitions rather than the sizes.

4.3 Exercise 2.2: relations in roster form

Question 3 uses A = {1, 2, 3, 5} and B = {4, 6, 9}, with a pair kept when its two entries have opposite parity. An odd minus an even, or an even minus an odd, gives an odd result, so exactly one entry of the pair must be odd. Checking the choices gives the pairs with 1, 2, 3 and 5 as first entries and a compatible second entry from B.

Question 5 defines R on A = {1, 2, 3, 4, 6} by the rule that b is exactly divisible by a. Start with a = 1 and keep every b, then a = 2 keeps 2, 4 and 6, then a = 3 keeps 3 and 6, a = 4 keeps 4, and a = 6 keeps 6. The roster is the sorted list of these pairs, the domain is all of A, and the range is also all of A because each element divides itself.

Question 8 asks for the number of relations from A = {x, y, z} to B = {1, 2}. The product has 3 times 2 = 6 pairs, so the number of relations is 2 to the power 6 = 64.

Question 9 defines R on the integers by the rule that a - b is an integer. Every pair of integers satisfies it, since the difference of two integers is always an integer. So the relation is the whole of Z x Z, its domain is Z and its range is Z.

4.4 Exercise 2.3: functions, domains and ranges

Question 1 tests whether given rosters are functions. The set {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} is a function: each first entry appears once, with image 1. The set {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)} is also a function. The set {(1, 3), (1, 5), (2, 5)} is not, because 1 has two images.

Question 2 asks for the domain and range of f(x) = -|x| and of the square root of 9 - x squared. For the first, the modulus gives non-negative outputs, and the minus sign flips them, so the range is all values at most zero while the domain is R. For the second, 9 - x squared must be at least zero, so x lies in [-3, 3], and the square root values fill [0, 3].

Question 4 converts Celsius to Fahrenheit using t(C) = 9C/5 + 32. Then t(0) = 32, t(28) = 82.4, t(-10) = 14, and t(C) = 212 gives C = 100.

Question 5 asks for ranges. For f(x) = 2 - 3x with x > 0, the outputs are less than 2, so the range is the open interval from minus infinity to 2. For f(x) = x squared + 2 the range is [2, infinity). For f(x) = x on the reals the range is R.

4.5 Miscellaneous exercise: algebra and domain questions

Question 7 uses f(x) = x + 1 and g(x) = 2x - 3. Then f + g has the rule 3x - 2, f - g has the rule 4 - x, and the quotient has the rule (x + 1)/(2x - 3) with x not equal to 3/2.

Question 6 asks for the range of f(x) = x squared / (1 + x squared). The denominator is always positive, and the value is 0 at x = 0. For large x the fraction approaches 1 but never reaches it, and the fraction cannot be negative. So the range is the half open interval [0, 1).

Question 12 defines f on A = {9, 10, 11, 12, 13} by taking the highest prime factor of n. Compute each: 9 gives 3, 10 gives 5, 11 gives 11, 12 gives 3 and 13 gives 13. The range is {3, 5, 11, 13}.

Question 8 asks for a linear function through the given points. If f(x) = ax + b and f(0) = -1, then b = -1. Using f(1) = 1 gives a - 1 = 1, so a = 2 and f(x) = 2x - 1. Check the third point to confirm.

Question 10 gives a relation with (2, 9) and (2, 11). It is a relation from A to B because every pair uses an element of A first and an element of B second. It is not a function because the first entry 2 has two images.

4.6 Common slips in these exercises

Three slips repeat every year. First, writing the product with bracketed pairs but losing the order, so (a, r) becomes (r, a). Second, treating the range as if it were the codomain, when the range only keeps images that actually occur. Third, declaring a relation a function without checking the whole domain set.

A fourth slip belongs to algebra of functions: forgetting to exclude the zeros of the denominator in a quotient. Write the excluded values beside the rule, not in a separate line, so the examiner sees them attached to the function.

Before you close an exercise, check the answer format. Roster form needs braces and commas, set-builder form needs the colon rule, and an interval needs the correct bracket at each end. The format carries marks even when the values are right.