Try our Free Online Math Solver!

Functions Inverse functions and composition
Course administration
Midterm:
• Wednesday, February 15, 2006
• Closed book, inclass
• 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 onetoone, 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 onetoone.
Alternate: A function is onetoone 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 codomain elements are covered
Bijective functions
Definition: A function f is called a bijection if it is both
onetoone
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 onetoone (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 onetoone (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 codomain is covered thus the
function is also a surjection (and bijection)
• A is finite and f is an onto function
• Is the function onetoone?
• Let f be a function from a set A to itself, that is
f: A>A
Assume
• A is finite and f is onetoone (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 codomain is covered thus the
function is also a surjection (and bijection)
• A is finite and f is an onto function
• Is the function onetoone?
• Yes. Every element maps to exactly one element and all elements
in A are covered. Thus the mapping must be onetoone
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 onetoone.
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 onetoone (injective).
Example:
• Let g : R → R, where g(x) = 2x  1. Is it increasing ?
• Proof .
For x>y holds 2x > 2y and subsequently 2x1 > 2y1
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 onetoone 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(n1) such that 1! = 1
Prev  Next 