Discrete Mathematics

ICS 141: Discrete Mathematics I (Fall 2014)
2.3 Functions
Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element
of B to each element of A. If f is a function from A to B, wee write f : A → B.
Domain, Codomain, Image, Preimage, Range
A function from A to B:
f :A→B
A is the domain
B is the codomain
a ∈ A, b ∈ B such that f (a) = b

a is the preimage of b under f
b is the image of a under f
The range is a specific subset of the Codomain(B) containing the actual values the function out-
puts.
The range can be written as f (A).
Injection (One-to-One)
A function where each element in the Domain maps to a single, unique element in the Codomain.
[Domain and Range have the same cardinality]. Strictly increasing or strictly decreasing functions
are one-to-one.
Surjection (Onto)
A function where every element in the Codomain is a valid output of the function. [Range is equal
to Codomain].
Bijection
A function that is both an injection and a surjection.
Identity Function
A function that maps f : A → A, such that f (a) = a where a ∈ A.
Inverse Function
Given the bijective function f , such that f : A → B and f (a) = b where a ∈ A and b ∈ B, the
inverse function is defined as f −1 , such that f −1 : B → A and f −1 (b) = a.
1
Composition
Given two functions, f and g, such that the range of g is a subset of the domain of f , the composi-
tion of f with g (f ◦ g) is defined as f (g(x)), with x ∈(g’s domain).
Floor Function
bxc returns the largest integer ≤ x.
Ceiling Function
dxe returns the smallest integer ≥ x.
2.3 pg 153 # 13
Determine whether each of these functions from Z to Z is onto (surjective).
a) f (n) = n − 1
This is surjective since every integer is 1 less than some integer.
b) f (n) = n2 + 1
Not surjective because the range cannot include negative integers.
c) f (n) = n3
Not surjective because any element in the codomain that is not a perfect cube will not be
mapped to.
2.3 pg 153 # 23
Determine the type of each function from R to R
a) f (x) = 2x + 1
Bijective. This is injective because for every a 6= b, we have f (a) 6= f (b) (every number is
1 more than 2 times some number). We also know that the function is surjective because the
range is all real numbers from 2((y − 1)/2) + 1 = y.
b) f (x) = x2 + 1
Not injective and not surjective. We know the function is not injective because we can have
the same value for f (x) given two different x values. For example, f (2) = 22 + 1 = 5 and
f (−2) = (−2)2 + 1 = 5. The function is also not surjective because the range is all real
numbers greater than or equal to 1, or can be written as [1, ∞).
c) f (x) = x3
Bijective. This is injective because for every a 6= b, we have f (a) 6= f (b) (every number is
the cube of some number). We also know that the function is surjectve because the range is
all real numbers from (y 1/3 )3 = y.
2
d) f (x) = (x2 + 1)/(x2 + 2)

Not injective and not surjective. We know the function is not injective because we can have
the same value for f (x) given two different x values. The function is also not surjective
because the range is only [0.5, 1).
Extra Problem
Given the following functions f and g, from R to R, find f ◦ g.
a) f (x) = x2
g(x) = x + 1
(f (g(x)) = f (x + 1) = (x + 1)2
b) f (x) = 2x + 1
g(x) = x2 + 4x + 4
(f (g(x)) = f (x2 + 4x + 4) = 2(x2 + 4x + 4) + 1 = 2x2 + 8x + 9
c) f (x) = {(1, 3), (2, 4), (5, 6), (4, 8)}
g(x) = {(1, 1), (4, 5), (6, 2)}
(f ◦ g) = {(1, 3), (4, 6), (6, 4)}
2.3 pg 154 # 31
Let f (x) = bx2 /3c. Find f (S) if

c) S = {1, 5, 7, 11}
f (1) = b12 /3c = b1/3c = 0
f (5) = b52 /3c = b25/3c = 8
f (7) = b72 /3c = b49/3c = 16
f (11) = b112 /3c = b121/3c = 40
Therefore, f (S) = {0, 8, 16, 40}
d) S = {2, 6, 10, 14}
f (2) = b22 /3c = b4/3c = 1
f (6) = b62 /3c = b36/3c = 12
f (10) = b102 /3c = b100/3c = 33
f (14) = b142 /3c = b196/3c = 65
Therefore, f (S) = {1, 12, 33, 65}
2.3 pg 154 # 43
Let g(x) = bxc. Find

a) g −1 ({0})
We need to find the set of all numbers whose floor is 0. Since all number from 0 to 1
(including 0 and excluding 1) round down to 0, then g −1 ({0}) = {x | 0 ≤ x < 1}
3
b) g −1 ({−1, 0, 1})
We know that the numbers from -1 to 2 (exclusive) round down to either -1, 0, or 1, then
g −1 ({−1, 0, 1}) = {x | −1 ≤ x < 2}
c) g −1 ({x | 0 < x < 1})

Since g(x) = bxc will always result in an integer, no value of x will result in a number
between 0 and 1. Thus, the image of the inverse function is the empty set, ∅
2.3 pg 155 # 69
Find the inverse function of f (x) = x3 + 1.
Solve for x.
y = x3 + 1
y − 1 = x3
(y − 1)1/3 = x
The inverse function function is f −1 (x) = (x − 1)1/3 .
Extra Problem
For each function from R to R, if the function has a defined inverse, find it.
a) f (x) = x2 − 2
This function is not bijective, so there is no inverse function.
b) f (x) = 3
This function is not bijective, so there is no inverse function.

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ICS 141: Discrete Mathematics I (Fall 2014)

Domain, Codomain, Image, Preimage, Range

a ∈ A, b ∈ B such that f (a) = b

A function that is both an injection and a surjection.

A function that maps f : A → A, such that f (a) = a where a ∈ A.

bxc returns the largest integer ≤ x.

dxe returns the smallest integer ≥ x.

Determine whether each of these functions from Z to Z is onto (surjective).

Determine the type of each function from R to R

d) f (x) = (x2 + 1)/(x2 + 2)

Given the following functions f and g, from R to R, find f ◦ g.

Let f (x) = bx2 /3c. Find f (S) if

Let g(x) = bxc. Find

c) g −1 ({x | 0 < x < 1})

Find the inverse function of f (x) = x3 + 1.

The inverse function function is f −1 (x) = (x − 1)1/3 .

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