4. Determine whether each of these functions from {a, b, c, d} to itself is one-to-one (onto)a) f (a) = b, f (b) = a, f (c) = c, f (d) = db) f (a) = b, f (b) = b, f (c) = d, f (d) = cc) f (a) = d, f (b) = b, f (c) = c, f (d) = d

5. Determine whether each of these functions from Z to Z is one-to-one (onto)a) f (n) = n − 1 b) f (n) = n2 + 1 c) f (n) = n3 d) 2nf n     6. Determine whether f : Z × Z → Z is onto ifa) f (m, n) = 2m − n b) f (m, n) = m2 − n2 c) f (m, n) = m + n + 1d) f (m, n) = |m| − |n| e) f (m, n) = m2 − 4 f) f (m, n) = m + n7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 18. Let S = {−1, 0, 2, 4, 7}. Find f (S) ifa) f (x) = 1 b) f (x) = 2x + 1 c) 5xf x     

7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 1

Let X, Y and Z be any non-empty sets and let f and g be one-one functions of X onto Y andY onto Z respectively so that f and g are both invertible. Then, show that(a) g ◦ f is one-one(b) g ◦ f is onto(c) (g ◦ f )−1 = f −1 ◦g−1

Prove.(a) f : Z+ → Z+, f (x) = 2x + 1 is one-to-one.(b) f : Z+ → Z, f (x) = 2x − x2 is not one to one.(c) f : R \ {0} → R+, f (x) = 1x2 is onto.(d) f : Z+ → Z+, f (x) = 2x − 1 is not onto.

Question 1 of 10Which of the following pairs of functions are inverses of each other?A. and B. and C. and D. and