Relations and Functions — Practice

Question 1

If A = {1, 2} and B = {a, b, c}, find A x B and state n(A x B).

Accepted Answer

A x B = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}. n(A x B) = n(A) x n(B) = 2 x 3 = 6.

Question 2

If A x B has 12 elements and A = {1,2,3,4}, find n(B) and give one possible set B.

Accepted Answer

n(A x B) = n(A) x n(B), so 12 = 4 x n(B), giving n(B) = 3. One possible B is {5,6,7} -- any 3-element set works, since only the size is determined.

Question 3

Let A = {1,2,3,4,5}. Define R = {(x,y) : y = x+2, x,y in A}. Write R in roster form and state its domain and range.

Accepted Answer

Checking x=1,2,3 gives y=3,4,5 (all in A); x=4,5 would need y=6,7, which are not in A. R={(1,3),(2,4),(3,5)}. Domain={1,2,3}. Range={3,4,5}.

Question 4

Let A = {2,3,4,5,6}. Define R on A by R = {(a,b) : a,b in A, a+b=8}. Write R in roster form, and find its domain and range.

Accepted Answer

Pairs from A summing to 8: (2,6),(3,5),(4,4),(5,3),(6,2). R={(2,6),(3,5),(4,4),(5,3),(6,2)}. Domain={2,3,4,5,6}. Range={2,3,4,5,6}.

Question 5

Is {(1,4),(2,4),(3,4),(4,4)} a function? Is {(4,1),(4,2),(5,3)} a function? Justify each.

Accepted Answer

The first IS a function: every distinct first element (1,2,3,4) has exactly one image (all happen to be 4, which is fine -- many inputs sharing one output is still allowed). The second is NOT a function: the first element 4 appears twice, pointing to both 1 and 2.

Question 6

State the domain and range of (i) f(x) = |x-3|, (ii) f(x) = 1/(x-2), (iii) f(x) = [x] (greatest integer function).

Accepted Answer

(i) Domain R, range [0,infinity), since an absolute value is never negative. (ii) Domain R minus {2} (denominator zero at x=2), range R minus {0} (this quotient can never actually equal zero). (iii) Domain R, range Z (the integers), since the output is always some whole number.

Question 7

State the domain and range of f(x) = 3^x and g(x) = log_3(x), and identify one point each graph must pass through.

Accepted Answer

f(x)=3^x: domain R, range (0,infinity), passes through (0,1) since 3^0=1. g(x)=log_3(x): domain (0,infinity), range R, passes through (1,0) since log_3(1)=0. Each function's domain is the other's range, since they are mirror images of each other.

Question 8

Let f(x) = x^2 - 1 and g(x) = x + 1, both defined on R. Find (f+g)(x), (f-g)(x), (fg)(x), and (f/g)(x), stating any domain restriction.

Accepted Answer

(f+g)(x) = x^2+x. (f-g)(x) = x^2-x-2. (fg)(x) = (x^2-1)(x+1) = x^3+x^2-x-1. (f/g)(x) = (x^2-1)/(x+1) = (x-1)(x+1)/(x+1) = x-1, valid for all x except x=-1, where the original quotient is undefined even though it simplifies algebraically.

Question 9

R is a relation on Z defined by R = {(a,b) : a,b in Z, a-b is divisible by 3}. Is (5,2) in R? Is (5,1) in R? Find one more pair with first element 5.

Accepted Answer

5-2=3, divisible by 3, so (5,2) IS in R. 5-1=4, not divisible by 3, so (5,1) is NOT in R. Another valid pair: (5,8), since 5-8=-3, which is divisible by 3.

Practice — Relations and Functions

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