# Functions Inverse functions and composition

## Course administration

**Midterm:**

• Wednesday, February 15, 2006

• Closed book, in-class

• Covers Chapter 1 of the textbook

**Homework 4 is out**

• Due on Friday, February 10, 2006

**• Definition:** Let A and B be two sets. A **function from A to B,**

denoted **f:** **A → B** , is an assignment of exactly one element of

B to each element of A. We write f(a) = b to denote the

assignment of b to an element a of A by the function f.

**Injective function**

**Definition:** A function f is said to be **one-to-one, or injective**,
if

and only if f(x) = f(y) implies x = y for all x, y in the domain of

f. A function is said to be an **injection if it is one-to-one.**

**Alternate:** A function is one-to-one if and only if f(x) ≠ f(y),

whenever x ≠ y. This is the contrapositive of the definition.

**Surjective function**

**Definition:** A function f from A to B is called **onto, or surjective,**

if and only if for every b ∈ B there is an element a ∈ A such that

f(a) = b.

**Alternative:** all co-domain elements are covered

**Bijective functions**

**Definition:** A function f is called **a bijection **if it is **both
one-to-one
and onto.**

• Let f be a function from a set A to itself, that is

f: A->A

**Assume**

**• A is finite and f is one-to-one (injective)
**• Is f an onto function (surjection)?

• Let f be a function from a set A to itself, that is

f: A->A

**Assume**

**• A is finite and f is one-to-one (injective)**

• Is f an onto function (surjection)?

**• Yes.** Every element points to exactly one element. Injection

assures they are different . So we have |A| different elements A

points to. Since f:A->A the co-domain is covered thus the

function is also a surjection (and bijection)

**• A is finite and f is an onto function**

• Is the function one-to-one?

• Let f be a function from a set A to itself, that is

f: A->A

**Assume**

**• A is finite and f is one-to-one (injective)**

• Is it an onto function (surjection)?

• Yes. Every element points to exactly one element. Injection

assures they are different . So we have |A| different elements A

points to. Since f: A -> A the co-domain is covered thus the

function is also a surjection (and bijection)

• A is finite and f is an onto function

**• Is the function one-to-one?**

**• Yes.** Every element maps to exactly one element and all elements

in A are covered. Thus the mapping must be one-to-one

**Functions on real numbers **

**Definition: **Let f1 and f2 be functions from A to R (reals). Then

f1 + f2 and f1 * f2 are also functions from A to R defined by

**Examples:**

• Assume

• f1(x) = x - 1

• f2(x) = x3 + 1

then

• (f1 + f2)(x) = x3 + x

• (f1 * f2)(x) = x4 - x3 + x - 1.

**Increasing and decreasing functions**

**Definition:** A function f whose domain and codomain
are subsets

of real numbers is **strictly increasing** if f(x) > f(y) whenever x >

y and x and y are in the domain of f. Similarly, f is called

**strictly decreasing** if f(x) < f(y) whenever x > y and x and y are

in the domain of f.

**Note:** Strictly increasing and strictly decreasing
functions are one-to-one.

**Example:**

• Let g : R → R, where g(x) = 2x - 1. Is it increasing ?

**Definition:** A function f whose domain and codomain are subsets

of real numbers is **strictly increasing** if f(x) > f(y) whenever x >

y and x and y are in the domain of f. Similarly, f is called

**strictly decreasing **if f(x) < f(y) whenever x > y and x and y are

in the domain of f.

**Note:** Strictly increasing and strictly decreasing functions are
one-to-one (injective).

**Example:**

• Let g : R → R, where g(x) = 2x - 1. Is it increasing ?

**• Proof .**

**For **x>y holds 2x > 2y and subsequently 2x-1 > 2y-1

**Thus g is strictly increasing.**

**Definition: **Let A be a set. The **identity function** on A is the

function i_{A}: A → A where i_{A}(x) = x.

**Example:**

• Let A = {1,2,3}

**Then:**

• i_{A}(1) = ?

**Definition:** Let A be a set. The** identity function** on A is the

function i_{A}: A → A where i_{A}(x) = x.

**Example:**

• Let A = {1,2,3}

**Then:**

• i_{A}(1) = 1

• i_{A}(2) = ?

**Definition:** Let A be a set. The **identity function **on A is the

function i_{A}: A → A where i_{A}(x) = x.

**Example:**

• Let A = {1,2,3}

**Then:**

• i_{A}(1) = 1

• i_{A}(2) = 2

• i_{A}(3) = 3.

**Bijective functions**

**Definition:** A function f is called **a bijection** if it is **both**
**one-to-one and onto.**

**Inverse functions**

**Definition:** Let f be a bijection from set A to set B. The **inverse
function of f **is the function that assigns to an element b from B

the unique element a in A such that f(a) = b. The inverse

function of f is denoted by f

^{-1}. Hence, f

^{-1}(b) = a, when f(a) = b.

If the inverse function of f exists, f is called

**invertible.**

Note: if f is not a bijection then it is not possible to define the

inverse function of f.

**Definition:** Let f be a bijection from set A to set B. The **inverse
function of f** is the function that assigns to an element b from B

the unique element a in A such that f(a) = b. The inverse

function of f is denoted by f

^{-1}. Hence, f

^{-1}(b) = a, when f(a) = b.

If the inverse function of f exists, f is called

**invertible.**

Note: if f is not a bijection then it is not possible to define the

inverse function of f.

**Example 1:**

• Let A = {1,2,3} and i_{A} be the identity function

• Therefore, the inverse function of i_{A } is i_{A}.

**Inverse functions**

**Example 2:**

• Let g : R → R, where g(x) = 2x - 1.

• What is the inverse function g^{-1} ?

**Approach to determine the inverse:**

y = 2x - 1 => y + 1 = 2x

=> (y+1)/2 = x

• Define g^{-1}(y) = x= (y+1)/2

**Test the correctness of inverse:**

• g**(3)** = ..

**Example 2:**

• Let g : R → R, where g(x) = 2x - 1.

• What is the inverse function g^{-1} ?

**Approach to determine the inverse:**

y = 2x - 1 => y + 1 = 2x

=> (y+1)/2 = x

• Define g^{-1}(y) = x= (y+1)/2

**Test the correctness of inverse:**

• g(3) = 2*3 - 1 = 5

• g^{-1} (5) =

**Example 2:**

• Let g : R → R, where g(x) = 2x - 1.

• What is the inverse function g^{-1} ?

**Approach to determine the inverse:**

y = 2x - 1 => y + 1 = 2x

=> (y+1)/2 = x

• Define g^{-1}(y) = x= (y+1)/2

**Test the correctness of inverse:**

• g(3) = 2*3 - 1 = 5

• g^{-1} (5) = (5+1)/2 = 3

• g(10) =

**Example 2:**

• Let g : R → R, where g(x) = 2x - 1.

• What is the inverse function g^{-1} ?

**Approach to determine the inverse:**

y = 2x - 1 => y + 1 = 2x

=> (y+1)/2 = x

• Define g^{-1}(y) = x= (y+1)/2

**Test the correctness of inverse:**

• g(3) = 2*3 - 1 = 5

• g^{-1} (5) = (5+1)/2 = 3

• g(10) = 2*10 - 1 = 19

• g^{-1} (19) =

**Example 2:**

• Let g : R → R, where g(x) = 2x - 1.

• What is the inverse function g^{-1} ?

**Approach to determine the inverse:**

y = 2x - 1 => y + 1 = 2x

=> (y+1)/2 = x

• Define g^{-1}(y) = x= (y+1)/2

**Test the correctness of inverse:**

• g(3) = 2*3 - 1 = 5

• g^{-1} (5) = (5+1)/2 = 3

• g(10) = 2*10 - 1 = 19

• g^{-1} (19) = (19+1)/2 = 10.

**Composition of functions**

**Definition:** Let f be a function from set A to set B and let g be a

function from set B to set C. The **composition of the functions
g and f**, denoted by g f is defined by

**Example 1:**

• Let A = {1,2,3} and B = {a,b,c,d}

**Example 1:**

• Let A = {1,2,3} and B = {a,b,c,d}

**Example 1:**

• Let A = {1,2,3} and B = {a,b,c,d}

**Example 1:**

• Let A = {1,2,3} and B = {a,b,c,d}

**Example 2:**

• Let f and g be function from Z into Z, where

• f(x) = 2x and g(x) = x^2

**Example 3:**

• (f f^{-1} )(x) = x and (f^{-1} f)(x) = x, for all x.

• Let f : R → R, where f(x) = 2x – 1 and f^{-1} (x) = (x+1)/2.

**Some functions**

**Definitions:**

• The** floor function** assigns a real number x the largest integer

that is less than or equal to x. The floor function is denoted by

• The **ceiling function** assigns to the real number x the smallest

integer that is greater than or equal to x. The ceiling function is

denoted by

Other important functions:

• Factorials : n! = n(n-1) such that 1! = 1

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