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1. INTRODUCTORY CALCULUS
ICTE115
OSAFO APEANTI WILSON

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

3. Example

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.

12. Vertical Line Test
• Functions

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

21. Evaluating Functions: Negatives

22. Evaluating Functions: Quotients

23. Evaluating the Difference Quotient

24. Evaluating the Difference Quotient

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

28. Common Functions

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?

40. INCREASING, DECREASING, AND
CONSTANT FUNCTIONS

41. INCREASING, DECREASING, AND
CONSTANT FUNCTIONS

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