SlideShare a Scribd company logo
INTRODUCTORY CALCULUS
ICTE115
OSAFO APEANTI WILSON
Relations and Functions
A relation between two sets A and B is a collection of ordered pairs, where the
first coordinate comes from A (domain) and the second comes from B (range).
E.g. if A = {1, 2, 3, 4} and B = {a, b, c}, one relation is the three pairs {(1, c), (1,a),
(3, a)}.
A function on sets A and B is a special kind of relation where every element of A
(domain) is paired with exactly one element from B (range).
e.g. {(1, a), (2, a), (3, b), (4, b)} is a function from A to B.
The relation above fails to be a function in two ways.
• Not every element of A is paired with an element from B
• The element 1 is used twice, not once.
Note there is no such restrictions on B; that is, elements from B can be paired
with elements from A many times or not at all.
1
2
3
4
a
b
c
A B
1
2
3
4
a
b
c
A B
Example
Examples
Consider the following three relations on the set A = {1, 2, 3}:
• f = {(1, 3), (2, 3), (3, 1)},
• g= {(1, 2), (3, 1)},
• h= {(1, 3), (2, 1), (1, 2), (3, 1)}
f is a function from A into A since each member of A appears
as the first coordinate in exactly one ordered pair in f; here
f (1) = 3, f (2) = 3, and f (3) = 1.
g is not a function from A into A since 2 ∈ A is not the
first coordinate of any pair in g and so g does not assign any
image to 2.
h is not a function from A into A since 1 ∈ A appears as the
first coordinate of two distinct ordered pairs in h, (1, 3) and
(1, 2). If h is to be a function it cannot assign both 3 and 2
to the element 1 ∈ A.
Relations and Functions
Try
• Let X = {1, 2, 3, 4}. Determine whether each
relation on X is a function from X into X.
(a) f = {(2, 3), (1, 4), (2, 1), (3.2), (4, 4)}
(b) g = {(3, 1), (4, 2), (1, 1)}
(c) h = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}
Relations and Functions
DEFINITION
• A function ƒ from a set A to a set B is a rule that assigns a unique
(single) element ƒ(x) ∈ B to each element x ∈ A.
• The set A of all possible input values is called the domain of the function.
• The set B is called the target set or co-domain.
• The set of all values of ƒ(x) as x varies through out A is called the range of the
function.
• The range may not include every element in the set B (co-domain).
1
2
3
4
a
b
c
A (domain) B (Co-domain)
Range
Functions and Algorithms
let f denote a function from A into B. Then we
write f: A → B
• If 2 ∈ 𝐴, then 𝑓 (2) (read: “f of 2”) denotes
the unique element of B which f assigns to a; it
is called the image of 2 under f, or the value of f
at 2. in this case a
1
2
3
4
a
b
c
A B
f: A → B
Functions Defined by Equations
• Consider the equation y=x2 - 3x, where x can be any
real number. This equation assigns to each x-value
exactly one corresponding y- value.
• Example
Functions Defined by Equations
• Note that y depends on what value of x is
selected,
• y is the dependent variable (Range) .
• x is the independent variable (Domain)
• it is important to recognize that not all
equations define functions.
Functions Defined by Equations
• An equation is a function if each element in
the domain corresponds to exactly one
element in the range.
• Example
Not FunctionsFunctions
Vertical Line Test
• Given the graph of an equation, if any vertical
line that can be drawn intersects the graph at
no more than one point, the equation defines
a function of x.
• This test is called the vertical line test.
Vertical Line Test
• Functions
Vertical Line Test
• Not functions
Expressing a Function
• A function can be expressed one of four ways:
– verbally,
– numerically
– algebraically
– graphically
This is sometimes called the Rule of 4.
Function Notation
• Equation y = 2x + 5 is a function
• If we give the function a name, say, “ƒ ”, then we can
use function notation:
f (x) = 2x + 5
The symbol f (x) is read “f evaluated at x” or “f of x” and
represents the y-value that corresponds to a particular
x-value. In other words, y = f (x) a function can be
expressed by means of a mathematical formula.
The function 𝑓 𝑥 = 2𝑥 + 5 can be writen as 𝑥 →
2𝑥 + 5 or 𝑦 = 2𝑥 + 5
Function Notation
It is important to note that f is the function name,
whereas f (x) is the value of the function
Evaluating Functions by Substitution
Evaluating Functions by Substitution
Find 𝑓 (−2), 𝑓 (0), and 𝑓 (6) for 𝑓 (𝑥) = 𝑥 + 3.
We need to substitute −2, 0, and 6 for x in the function.
𝑓 (−2) = −2 + 3 = √1 = 1
𝑓 (0) = 0 + 3 = √3
𝑓 (6) = 6 + 3 = √9 = 3
Find 𝑓 (−8), 𝑓 (𝜋), and 𝑓 (10) for 𝑓 (𝑥) = 16.
𝑓 (𝑥) = 16 is a constant function, so the y-value is 16 no matter what quantity
is in the parentheses.
𝑓 (−8) = 16 𝑓 (𝜋) = 16 𝑓 (10) = 16
Evaluating Functions by Substitution
A piecewise function is a function with two or more formulas for computing 𝑦.
The formula to use depends on where 𝑥 is. There will be an interval for
𝑥 written next to each formula for 𝑦.
𝑓 𝑥 =
𝑥 − 1 𝑖𝑓 𝑥 ≤ −2
2𝑥 𝑖𝑓 − 2 < 𝑥 < 2
𝑥2
𝑖𝑓 𝑥 ≥ 2
In this example, there are three formulas for 𝑦: 𝑦 = 𝑥 − 1, 𝑦 = 2𝑥, and
𝑦 = 𝑥2, and three intervals for 𝑥: 𝑥 ≤ −2, −2 < 𝑥 < 2, and 𝑥 ≥ 2.
When evaluating this function, we need to decide to which interval 𝑥 belongs.
Then we will use the corresponding formula for 𝑦.
Evaluating Functions by Substitution
To find f (5), f (−3), and 𝑓 (0) for the function above.
For 𝑓 (5), does 𝑥 = 5 belong to 𝑥 ≤ −2, −2 < 𝑥 < 2, or 𝑥 ≥ 2?
Because 5 ≥ 2, we will use 𝑦 = 𝑥2, the formula written next to 𝑥 ≥ 2.
𝑓 (5) = 52 = 25
For 𝑓 (−3), does 𝑥 = −3 belong to 𝑥 ≤ −2, −2 < 𝑥 < 2, 𝑜𝑟 𝑥 ≥ 2?
Because −3 ≤ −2, we will use 𝑦 = 𝑥 − 1, the formula written next to
𝑥 ≤ −2.
𝑓 (−3) = −3 − 1 = −4
For 𝑓 (0), does 𝑥 = 0 belong to 𝑥 ≤ −2, −2 < 𝑥 < 2, 𝑜𝑟 𝑥 ≥ 2?
Because
−2 < 0 < 2, we will use 𝑦 = 2𝑥, the formula written next to −2 <
𝑥 < 2.
𝑓 (0) = 2(0) = 0
Evaluating Functions: Negatives
Evaluating Functions: Quotients
Evaluating the Difference Quotient
Evaluating the Difference Quotient
Introductory calculus
Domain of a Function
• “What can x be?” The domain of a function excludes
values that cause a function to be undefined or have
outputs that are not real numbers.
Determining the Domain of a
Function
Common Functions
Introductory calculus
Introductory calculus
Introductory calculus
Introductory calculus
Introductory calculus
Even and Odd Functions
Determining Whether a Function Is
Even, Odd, or Neither
Determining Whether a Function Is
Even, Odd, or Neither
ONE-TO-ONE, ONTO, AND INVERTIBLE
FUNCTIONS
A function f : A → B is said to be one-to-one (written 1-1) if different
elements in the domain A have
distinct images. That is f is one-to-one if f (a) = f (a’) implies a = a’
A function f: A → B is said to be an onto function if each element of B
is the image of some element of A.
Thus: f : A → B is onto if the image of f is the entire codomain, i.e.,
if f (A) = B. In such a case we say that f is a function from A onto B or
that f maps A onto B.
A function f: A → B is invertible if its inverse relation f −1 is a function
from B to A. In general, the inverse
relation f −1 may not be a function.
Theorem : A function f: A → B is invertible if and only if f is both one-
to-one and onto.
EXAMPLE
Consider the functions f1: A → B, f2: B → C, f3: C → D
and f4: D → E defined by the diagram of below.
Which of the function(s) are
a) One-to-one
b) Onto
c) Invertible
Geometrical Characterization of One-
to-One and Onto Functions
(1) f :R → R is one-to-one if each horizontal line
intersects the graph of f in at most one point.
(2) f :R → R is an onto function if each horizontal line
intersects the graph of f at one or more points.
Which of the following function is one to one or onto?
INCREASING, DECREASING, AND
CONSTANT FUNCTIONS
INCREASING, DECREASING, AND
CONSTANT FUNCTIONS
AVERAGE RATE OF CHANGE
• The slope of the secant line is used to
represent the average rate of change of the
function.
AVERAGE RATE OF CHANGE
Introductory calculus
Introductory calculus
Introductory calculus

More Related Content

PPTX
Function and graphs
PDF
2.3 Functions
PPTX
Functions
PPT
4.6 Relations And Functions
PPT
Relations and functions
PPTX
Functions
PPTX
Relations & Functions
PPTX
The Algebric Functions
Function and graphs
2.3 Functions
Functions
4.6 Relations And Functions
Relations and functions
Functions
Relations & Functions
The Algebric Functions

What's hot (18)

PPS
Functions and graphs
PPT
7_Intro_to_Functions
PPTX
Relations & functions
PPTX
function
PPT
Functions
PPT
Lec 04 function
PPT
2.1 Functions and Their Graphs
PPTX
mathematical functions
PPTX
CBSE Class 12 Mathematics formulas
PPTX
Relations & functions.pps
PPTX
Function notation by sadiq
PPTX
Algebraic functions powerpoint
PPTX
7 functions
PPTX
Relations and functions
PPT
PPt on Functions
PPT
Relations and Functions
PPTX
PPT
Holt alg1 ch5 1 identify linear functions
Functions and graphs
7_Intro_to_Functions
Relations & functions
function
Functions
Lec 04 function
2.1 Functions and Their Graphs
mathematical functions
CBSE Class 12 Mathematics formulas
Relations & functions.pps
Function notation by sadiq
Algebraic functions powerpoint
7 functions
Relations and functions
PPt on Functions
Relations and Functions
Holt alg1 ch5 1 identify linear functions
Ad

Similar to Introductory calculus (20)

PDF
3.1 Functions and Function Notation
PPTX
CM 01 CP 02 Topic 03 Concept of Function.pptx
PPTX
Lec 11 Functions of discrete structure .pptx
PPTX
CMSC 56 | Lecture 9: Functions Representations
PPTX
9-Functions.pptx
PDF
functions-1.pdf
PPT
Relations and Functions
PPT
AXSARFERHBYUJKIOPOOIU7URTGERFEWRFDSFVDGREYGTH
PPTX
Functions
PPT
PPT
Functions (1)
PPTX
Relations and Functions – Understanding the Foundation of Mathematics.pptx
PPT
Functions
PPTX
Functions and it's graph6519105021465481791.pptx
PPT
Applied Calculus New Free Lecture 2.1 .ppt
PPTX
LCV-MATH-1.pptx
PPT
Calculus - 1 Functions, domain and range
PPTX
17.5 introduction to functions
PPTX
Lesson 1_Functions.pptx
PPTX
Functions it's types and relations etc...
3.1 Functions and Function Notation
CM 01 CP 02 Topic 03 Concept of Function.pptx
Lec 11 Functions of discrete structure .pptx
CMSC 56 | Lecture 9: Functions Representations
9-Functions.pptx
functions-1.pdf
Relations and Functions
AXSARFERHBYUJKIOPOOIU7URTGERFEWRFDSFVDGREYGTH
Functions
Functions (1)
Relations and Functions – Understanding the Foundation of Mathematics.pptx
Functions
Functions and it's graph6519105021465481791.pptx
Applied Calculus New Free Lecture 2.1 .ppt
LCV-MATH-1.pptx
Calculus - 1 Functions, domain and range
17.5 introduction to functions
Lesson 1_Functions.pptx
Functions it's types and relations etc...
Ad

Recently uploaded (20)

PDF
Microbial disease of the cardiovascular and lymphatic systems
PDF
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
PPTX
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
PPTX
GDM (1) (1).pptx small presentation for students
PDF
O7-L3 Supply Chain Operations - ICLT Program
PDF
Insiders guide to clinical Medicine.pdf
PDF
Anesthesia in Laparoscopic Surgery in India
PDF
ANTIBIOTICS.pptx.pdf………………… xxxxxxxxxxxxx
PDF
102 student loan defaulters named and shamed – Is someone you know on the list?
PPTX
human mycosis Human fungal infections are called human mycosis..pptx
PDF
VCE English Exam - Section C Student Revision Booklet
PDF
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
PDF
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH 9 GLOBAL SUCCESS - CẢ NĂM - BÁM SÁT FORM Đ...
PDF
grade 11-chemistry_fetena_net_5883.pdf teacher guide for all student
PPTX
Institutional Correction lecture only . . .
PDF
Chapter 2 Heredity, Prenatal Development, and Birth.pdf
PPTX
1st Inaugural Professorial Lecture held on 19th February 2020 (Governance and...
PPTX
Pharmacology of Heart Failure /Pharmacotherapy of CHF
PPTX
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
PPTX
IMMUNITY IMMUNITY refers to protection against infection, and the immune syst...
Microbial disease of the cardiovascular and lymphatic systems
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
GDM (1) (1).pptx small presentation for students
O7-L3 Supply Chain Operations - ICLT Program
Insiders guide to clinical Medicine.pdf
Anesthesia in Laparoscopic Surgery in India
ANTIBIOTICS.pptx.pdf………………… xxxxxxxxxxxxx
102 student loan defaulters named and shamed – Is someone you know on the list?
human mycosis Human fungal infections are called human mycosis..pptx
VCE English Exam - Section C Student Revision Booklet
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH 9 GLOBAL SUCCESS - CẢ NĂM - BÁM SÁT FORM Đ...
grade 11-chemistry_fetena_net_5883.pdf teacher guide for all student
Institutional Correction lecture only . . .
Chapter 2 Heredity, Prenatal Development, and Birth.pdf
1st Inaugural Professorial Lecture held on 19th February 2020 (Governance and...
Pharmacology of Heart Failure /Pharmacotherapy of CHF
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
IMMUNITY IMMUNITY refers to protection against infection, and the immune syst...

Introductory calculus

  • 2. Relations and Functions A relation between two sets A and B is a collection of ordered pairs, where the first coordinate comes from A (domain) and the second comes from B (range). E.g. if A = {1, 2, 3, 4} and B = {a, b, c}, one relation is the three pairs {(1, c), (1,a), (3, a)}. A function on sets A and B is a special kind of relation where every element of A (domain) is paired with exactly one element from B (range). e.g. {(1, a), (2, a), (3, b), (4, b)} is a function from A to B. The relation above fails to be a function in two ways. • Not every element of A is paired with an element from B • The element 1 is used twice, not once. Note there is no such restrictions on B; that is, elements from B can be paired with elements from A many times or not at all. 1 2 3 4 a b c A B 1 2 3 4 a b c A B
  • 4. Examples Consider the following three relations on the set A = {1, 2, 3}: • f = {(1, 3), (2, 3), (3, 1)}, • g= {(1, 2), (3, 1)}, • h= {(1, 3), (2, 1), (1, 2), (3, 1)} f is a function from A into A since each member of A appears as the first coordinate in exactly one ordered pair in f; here f (1) = 3, f (2) = 3, and f (3) = 1. g is not a function from A into A since 2 ∈ A is not the first coordinate of any pair in g and so g does not assign any image to 2. h is not a function from A into A since 1 ∈ A appears as the first coordinate of two distinct ordered pairs in h, (1, 3) and (1, 2). If h is to be a function it cannot assign both 3 and 2 to the element 1 ∈ A.
  • 5. Relations and Functions Try • Let X = {1, 2, 3, 4}. Determine whether each relation on X is a function from X into X. (a) f = {(2, 3), (1, 4), (2, 1), (3.2), (4, 4)} (b) g = {(3, 1), (4, 2), (1, 1)} (c) h = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}
  • 6. Relations and Functions DEFINITION • A function ƒ from a set A to a set B is a rule that assigns a unique (single) element ƒ(x) ∈ B to each element x ∈ A. • The set A of all possible input values is called the domain of the function. • The set B is called the target set or co-domain. • The set of all values of ƒ(x) as x varies through out A is called the range of the function. • The range may not include every element in the set B (co-domain). 1 2 3 4 a b c A (domain) B (Co-domain) Range
  • 7. Functions and Algorithms let f denote a function from A into B. Then we write f: A → B • If 2 ∈ 𝐴, then 𝑓 (2) (read: “f of 2”) denotes the unique element of B which f assigns to a; it is called the image of 2 under f, or the value of f at 2. in this case a 1 2 3 4 a b c A B f: A → B
  • 8. Functions Defined by Equations • Consider the equation y=x2 - 3x, where x can be any real number. This equation assigns to each x-value exactly one corresponding y- value. • Example
  • 9. Functions Defined by Equations • Note that y depends on what value of x is selected, • y is the dependent variable (Range) . • x is the independent variable (Domain) • it is important to recognize that not all equations define functions.
  • 10. Functions Defined by Equations • An equation is a function if each element in the domain corresponds to exactly one element in the range. • Example Not FunctionsFunctions
  • 11. Vertical Line Test • Given the graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the equation defines a function of x. • This test is called the vertical line test.
  • 13. Vertical Line Test • Not functions
  • 14. Expressing a Function • A function can be expressed one of four ways: – verbally, – numerically – algebraically – graphically This is sometimes called the Rule of 4.
  • 15. Function Notation • Equation y = 2x + 5 is a function • If we give the function a name, say, “ƒ ”, then we can use function notation: f (x) = 2x + 5 The symbol f (x) is read “f evaluated at x” or “f of x” and represents the y-value that corresponds to a particular x-value. In other words, y = f (x) a function can be expressed by means of a mathematical formula. The function 𝑓 𝑥 = 2𝑥 + 5 can be writen as 𝑥 → 2𝑥 + 5 or 𝑦 = 2𝑥 + 5
  • 16. Function Notation It is important to note that f is the function name, whereas f (x) is the value of the function
  • 17. Evaluating Functions by Substitution
  • 18. Evaluating Functions by Substitution Find 𝑓 (−2), 𝑓 (0), and 𝑓 (6) for 𝑓 (𝑥) = 𝑥 + 3. We need to substitute −2, 0, and 6 for x in the function. 𝑓 (−2) = −2 + 3 = √1 = 1 𝑓 (0) = 0 + 3 = √3 𝑓 (6) = 6 + 3 = √9 = 3 Find 𝑓 (−8), 𝑓 (𝜋), and 𝑓 (10) for 𝑓 (𝑥) = 16. 𝑓 (𝑥) = 16 is a constant function, so the y-value is 16 no matter what quantity is in the parentheses. 𝑓 (−8) = 16 𝑓 (𝜋) = 16 𝑓 (10) = 16
  • 19. Evaluating Functions by Substitution A piecewise function is a function with two or more formulas for computing 𝑦. The formula to use depends on where 𝑥 is. There will be an interval for 𝑥 written next to each formula for 𝑦. 𝑓 𝑥 = 𝑥 − 1 𝑖𝑓 𝑥 ≤ −2 2𝑥 𝑖𝑓 − 2 < 𝑥 < 2 𝑥2 𝑖𝑓 𝑥 ≥ 2 In this example, there are three formulas for 𝑦: 𝑦 = 𝑥 − 1, 𝑦 = 2𝑥, and 𝑦 = 𝑥2, and three intervals for 𝑥: 𝑥 ≤ −2, −2 < 𝑥 < 2, and 𝑥 ≥ 2. When evaluating this function, we need to decide to which interval 𝑥 belongs. Then we will use the corresponding formula for 𝑦.
  • 20. Evaluating Functions by Substitution To find f (5), f (−3), and 𝑓 (0) for the function above. For 𝑓 (5), does 𝑥 = 5 belong to 𝑥 ≤ −2, −2 < 𝑥 < 2, or 𝑥 ≥ 2? Because 5 ≥ 2, we will use 𝑦 = 𝑥2, the formula written next to 𝑥 ≥ 2. 𝑓 (5) = 52 = 25 For 𝑓 (−3), does 𝑥 = −3 belong to 𝑥 ≤ −2, −2 < 𝑥 < 2, 𝑜𝑟 𝑥 ≥ 2? Because −3 ≤ −2, we will use 𝑦 = 𝑥 − 1, the formula written next to 𝑥 ≤ −2. 𝑓 (−3) = −3 − 1 = −4 For 𝑓 (0), does 𝑥 = 0 belong to 𝑥 ≤ −2, −2 < 𝑥 < 2, 𝑜𝑟 𝑥 ≥ 2? Because −2 < 0 < 2, we will use 𝑦 = 2𝑥, the formula written next to −2 < 𝑥 < 2. 𝑓 (0) = 2(0) = 0
  • 26. Domain of a Function • “What can x be?” The domain of a function excludes values that cause a function to be undefined or have outputs that are not real numbers.
  • 27. Determining the Domain of a Function
  • 34. Even and Odd Functions
  • 35. Determining Whether a Function Is Even, Odd, or Neither
  • 36. Determining Whether a Function Is Even, Odd, or Neither
  • 37. ONE-TO-ONE, ONTO, AND INVERTIBLE FUNCTIONS A function f : A → B is said to be one-to-one (written 1-1) if different elements in the domain A have distinct images. That is f is one-to-one if f (a) = f (a’) implies a = a’ A function f: A → B is said to be an onto function if each element of B is the image of some element of A. Thus: f : A → B is onto if the image of f is the entire codomain, i.e., if f (A) = B. In such a case we say that f is a function from A onto B or that f maps A onto B. A function f: A → B is invertible if its inverse relation f −1 is a function from B to A. In general, the inverse relation f −1 may not be a function. Theorem : A function f: A → B is invertible if and only if f is both one- to-one and onto.
  • 38. EXAMPLE Consider the functions f1: A → B, f2: B → C, f3: C → D and f4: D → E defined by the diagram of below. Which of the function(s) are a) One-to-one b) Onto c) Invertible
  • 39. Geometrical Characterization of One- to-One and Onto Functions (1) f :R → R is one-to-one if each horizontal line intersects the graph of f in at most one point. (2) f :R → R is an onto function if each horizontal line intersects the graph of f at one or more points. Which of the following function is one to one or onto?
  • 42. AVERAGE RATE OF CHANGE • The slope of the secant line is used to represent the average rate of change of the function.
  • 43. AVERAGE RATE OF CHANGE