Discrete Mathematics Answers

Display(ii) f(x) = x
3 + 1 from the set of real numbers to the set of real numbers the graph of following functions.

Display the graph of following functions.
(i) f(n) = n
2 + 4 from the set of integers to the set of integers

(b) Given f(x) = x
2
, g(x) = 2x and h(x) = x - 2 be functions from R to R. Find
(i) f ◦ g
(ii) f ◦ h
(iii) h ◦ f
(iv) (f ◦ g) ◦ h
(v) f ◦ (g ◦ h)

(a) Consider f : R → R, f(x) = x
2
and g : R → R, g(x) = x + 5.
(i) What is g ◦ f?
(ii) What is f ◦ g

(iv) f : N → Y such that f(n) = 2n + 1,
where Y = {y ∈ N : y = 4n + 3 for some n ∈ N}

(b) Determine whether each of following functions are invertible. If it is an invertible, find
it’s inverse.
(i) f : R → R such that f(x) = 3x + 10
(ii) f : R → R such that f(x) = x
2
- 1

(a) Let A = {a,b,c,d,e,f} and B = {1,2,3,4,5,6}. Determine whether each of following functions from A to B are invertible. If it is an invertible, find it’s inverse.
(i) f1 = {(a,1), (b,2), (c,3), (d,4), (e,5), (f,6)}
(ii) f2 = {(a,2), (b,5), (d,2), (c,3), (e,5), (f,6)}
(iii) f3 = {(b,3), (d,6), (a,1), (c,3), (e,4), (f,5)}
(iv) f4 = {(a,6), (b,5), (c,1), (f,2), (d,4), (e,3)}

(b) Determine whether each of these function is a bijection
(i) f : N → N such that f(n) = n
2
(ii) f : N → N such that f(n) = n + 3
1
(iii) f : R → R such that f(x) = x
3
(iv) f : R → R such that f(x) = x
2 + 1
(v) f : N → N such that
f =
(
n − 1 : n is odd
n + 1 : n is even

Let g : R → R defined by the equation g(x) = x
2 + x. Let H ⊆ R and
H = {y ∈ R : 6 ≤ y ≤ 12}. Then determine the inverse image g
−1
(H).