1.
FUNCTIONS II
Dr. Gabriel Obed Fosu
Department of Mathematics
Kwame Nkrumah University of Science and Technology
Google Scholar: https://scholar.google.com/citations?user=ZJfCMyQAAAAJ&hl=en&oi=ao
ResearchGate ID: https://www.researchgate.net/profile/Gabriel_Fosu2
Dr. Gabby (KNUST-Maths) Functions 1 / 41

2.
Lecture Outline
1 Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
2 Inverse Function
3 Sequence and Series
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3.
Properties of Functions Odd and Even Functions
Even Function
Let f be a function and Df its domain. We assume that if x ∈ Df then −x ∈ Df .
Definition (Even Function)
f is an even function if f (−x) = f (x).
Example
The functions f (x) = x2
, g(x) = −x4
+ 2x2
− 1, h(x) = cos(x) + x2
, i(x) = x sinx are even
functions since:
1 f (−x) = (−x)2
= x2
= f (x),
2 g(−x) = −(−x)4
+2(−x)2
−1 = g(x),
3 h(−x) = cos(−x)+(−x)2
= h(x), and
4 i(−x) = (−x)sin(−x) = −x(−sinx) = x sinx = i(x).
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4.
Properties of Functions Odd and Even Functions
Odd Function
Definition (Odd Function)
f is an odd function if f (−x) = −f (x).
Example
The functions f (x) = x, f (x) = −x3
+ 2x, f (x) = sin(x), and f (x) = csc(x), f (x) = tan(x) are
odd functions. Because
1 f (−x) = −x = −f (x)
2 f (−x) = −(−x)3
+2(−x) = x3
−2(x) = −f (x)
3 f (−x) = sin(−x) = −sin(x) = −f (x)
4 f (−x) =
1
sin(−x)
= −
1
sin(x)
= −f (x)
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5.
Properties of Functions Odd and Even Functions
Remarks
The graph of an even function is symmetric about the y-axis.
The graph of an odd function is symmetric about the origin.
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6.
Properties of Functions Periodic Functions
Periodic Functions
Definition
Let f be a function, and Df its domain, then f is a periodic function if there exists a
positive real number t such that f (x + t) = f (x) for all x ∈ Df .
The minimum of such t’s which is often denoted as T , is called the period of f .
Example
The trigonometric functions are periodic functions.
1 sin(x +2kπ) = sin(x +2π) = sin(x) for k ∈ Z, however, T = 2π.
2 cos(x +2kπ) = cos(x +2π) = cos(x),T = 2π.
3 tan(x +(2k +1)π) = tan(x +π) = tan(x),T = π.
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7.
Properties of Functions Monotonic Functions
Monotonic Functions
Let I be an open interval. x1 and x2 are two elements of I such that x1 < x2.
Definition
☛ f is an increasing function on I if f (x1)<f (x2).
☛ f is a decreasing function on I if f (x1)>f (x2).
Example
• The functions ex
, tan(x) and ax + b, where a > 0, are increasing on their respective
domains.
• The functions e−x
, cot(x) and ax + b, where a < 0, are decreasing on their respective
domains.
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8.
Properties of Functions Monotonic Functions
Monotonic Decreasing Function
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
−3x +1 ↘
e−x
↘
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9.
Properties of Functions Monotonic Functions
Monotonic Increasing Function
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
2x −1 ↗
ex
↗
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10.
Properties of Functions Monotonic Functions
Monotonic Increasing and Decreasing
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
sinx ↗↘
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11.
Properties of Functions Monotonic Functions
Monotonic Functions
Example
Show that the function f (x) =
p
x −2 is an increasing function on its domain.
1 Df = [2,+∞).
2 For x1,x2 ∈ Df and x1 < x2,
3 2 < x1 < x2 =⇒ 0 < x1 −2 < x2 −2
4 =⇒ 0 <
p
x1 −2 <
p
x2 −2
5 =⇒ f (x1) < f (x2).
6 Thus, f is an increasing function on its domain.
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12.
Properties of Functions Monotonic Functions
Monotonic Functions
Example
Show that f (x) = (2− x)2
+1 decreases on (−∞,2] and increases on [2,+∞).
1 Df = R.
2 For x1,x2 ∈ (−∞,2],
3 x1 < x2 ≤ 2 =⇒ −x1 > −x2 > −2
4 =⇒ 2− x1 > 2− x2 > 0
5 =⇒ (2− x1)2
> (2− x2)2
> 0
6 =⇒ (2− x1)2
+1 > (2− x2)2
+1 > 1
7 =⇒ f (x1) > f (x2).
8 f is decreasing on (−∞,2].
1 For x1,x2 ∈ [2,+∞),
2 2 ≤ x1 < x2 =⇒ −2 > −x1 > −x2
3 =⇒ 0 > 2− x1 > 2− x2
4 =⇒ 0 < (2− x1)2
< (2− x2)2
5 =⇒ 1 < (2− x1)2
+1 < (2− x2)2
+1
6 =⇒ f (x1) < f (x2).
7 f is an increasing function on [2,+∞).
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13.
Properties of Functions Bounded Functions
Bounded Functions
Definition
A function is said to be bounded above if there is ū ∈ R such that f (x) ≤ ū for all x in the
domain of f .
Example
The function f (x) = x2
+ 1 defined on 0 ≤ x ≤ 1 is bounded above by 2 since f (x) ≤ 2 for
0 ≤ x ≤ 1.
Example
The function f (x) = 1/x defined on x ∈ N is bounded above by 1
Example
The function f (x) = sinx is bounded above by 1 for x ∈ R.
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14.
Properties of Functions Bounded Functions
Bounded Functions
Definition
A function, f , is said to be bounded below if there is ℓ ∈ R such that f (x) ≥ ℓ for all x in the
domain of f .
Example
The function f (x) = x−1 defined in [0,1] is bounded below by −1 since −1 ≤ f (x) for x ∈ [0,1].
Example
The function g(x) = |
p
x +1| is bounded below by 0 on the interval [0,4] since 0 ≤ g(x) for
x ∈ [0,4].
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15.
Properties of Functions Maxima and Minima of Functions
Maxima and Minima of Functions
Local(or relative) and Global(or absolute) Minimum
1 The function f is said to have a local minimum value at the point x0 if f (x0) ≤ f (x) for
all x in a neighbourhood of x0.
2 f is said to have a global minimum value at the point x0 if f (x0) ≤ f (x) for all x in the
domain of f .
3 In this case f is bounded below.
Local(or relative) and Global(or absolute) Maximum
1 If f (x) ≤ f (x0) for all x in a neighbourhood of x0, then f has a local maximum value at
the point x0.
2 The maximum is global if f (x) ≤ f (x0) for all x in the domain of f .
3 In this case f is bounded above.
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16.
Properties of Functions Maxima and Minima of Functions
Maxima and Minima of Functions
−5 −4 −2 2 4
−3
−1
1
2
3
0
global min
f (x) = 1+(x +1)2
,
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17.
Properties of Functions Maxima and Minima of Functions
Maxima and minima
−5 −4 −2 2 4
−3
−1
1
2
3
0
local max
local min
g(x) = 1−2x −3x2
+2x3
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18.
Properties of Functions Maxima and Minima of Functions
Maxima and minima
−5 −4 −2 2 4
−3
−1
1
2
3
0
global min
local max
local min
h(x) = (x −1)(−x +3)2
(x),
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19.
Inverse Function
Inverse Functions
1 An inverse function is a function that undoes the action of the another function.
2 A function g is the inverse of a function f if whenever y = f (x) then x = g(y)
3 In other words, applying f and then g is the same thing as doing nothing. We can
write this in terms of the composition of f and g as
g(f (x)) = x
4 A function f has an inverse function only if for every y in its range there is only one
value of x in its domain for which f (x) = y
5 This inverse function is unique and is frequently denoted by f −1
and called f inverse.
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20.
Inverse Function
Inverse Functions
Given the function f (x) we want to find the inverse function, f −1
(x)
1. First, replace f (x) with y.
2. Replace every x with a y and replace every y with an x.
3. Solve the equation from Step 2 for y. This is the step where mistakes are most often
made so be careful with this step.
4. Replace y with f −1
(x). In other words, we’ve managed to find the inverse at this point!
5. Verify your work by checking that (f ◦ f −1
)(x) = x and (f −1
◦ f )(x) = x are both true.
Example
Given f (x) = 3x −2 find f −1
(x)
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21.
Inverse Function
1 Given the function f
f (x) = 3x −2
2 First, replace f (x) with y.
y = 3x −2
3 Replace every x with a y and replace every y with an x.
x = 3y −2
4 Solve the equation from Step 2 for y.
y =
1
3
(x +2)
5 Replace y with f −1
(x).
f −1
(x) =
x
3
+
2
3
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22.
Inverse Function
We can verify the results, we check that (f ◦ f −1
)(x) = x
(f ◦ f −1
)(x) = f [f −1
(x)]
= f
hx
3
+
2
3
i
= 3
hx
3
+
2
3
i
−2
= x +2−2
= x
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23.
Inverse Function
Graph of a Function and Its Inverse
There is an interesting relationship between the graph of a function and the graph of its
inverse. Here is the graph of the function and inverse from the first two examples.
In both cases the graph of the inverse is a reflection of the actual function about the line
y = x . This will always be the case with the graphs of a function and its inverse.
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24.
Sequence and Series
Sequence and Series
Definition
A sequence is an ordered set of numbers that most often follows some rule (or pattern) to
determine the next term in the order.
Example
x,x2
,x3
,x4
, ... is a sequence of numbers, where each successive term is multiplied by x.
Definition
A series is a summation of the terms of a sequence. The greek letter sigma Σ is used to
represent the summation of terms of a sequence of numbers.
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25.
Sequence and Series
Sequence and Series
Series are typically written in the following form:
n
X
i=1
ai = a1 + a2 + a3 ···+ an
where the index of summation, i takes consecutive integer values from the lower limit, 1
to the upper limit, n. The term ai is known as the general term.
Example
5
X
i=1
i = 1+2+3+4+5 = 15 (1)
6
X
k=3
2k
= 23
+24
+25
+26
= 8+16+32+64 = 120 (2)
4
X
k=1
kk
= 11
+22
+33
+44
= 1+4+27+256 = 288 (3)
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26.
Sequence and Series
Finite Series
Definition (Finite and Infinite Series)
A finite series is a summation of a finite number of terms. An infinite series has an infinite
number of terms and an upper limit of infinity.
Properties of Finite Series
The following are the properties for addition/subtraction and scalar multiplication of series.
For some sequence ai and bi and a scalar k then:
n
X
i=1
kai ±bi = k
n
X
i=1
ai ±
n
X
i=1
bi
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27.
Sequence and Series
Theorems of Finite Series
1 The following theorems give formulas to calculate series with common general terms.
These formulas, along with the properties listed above, make it possible to solve any
series with a polynomial general term, as long as each individual term has a degree
of 3 or less.
n
X
i=1
1 = n (4)
n
X
i=1
c = nc (5)
n
X
i=1
i =
n(n +1)
2
(6)
n
X
i=1
i2
=
n(n +1)(2n +1)
6
(7)
n
X
i=1
i3
=
µ
n(n +1)
2
¶2
(8)
(9)
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28.
Sequence and Series
Types of Sequences
There are two main types of sequences.
1 An arithmetic sequence is one in which successive terms differ by the same amount.
For example, {3, 6, 9, 12, ···}. Note each term is obtained by adding 3 to the previous
term. This is called the common difference denoted as d = 6−3
2 A geometric sequence is one in which the quotient of any two successive terms is a
constant. For example, {3, 9, 27, 81, ···}. Note each term is obtained by multiplying
the previous term by 3. This is called the common ratio denoted by r = 9
3 .
Similarly, there are also arithmetic series and geometric series, which are simply
summations of arithmetic and geometric sequences, respectively.
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29.
Sequence and Series
Theorems for Arithmetic and Geometric Series
Arithmetic Series
Suppose we have the following arithmetic series,
{a +(a +d)+(a +2d)+···+(a +(n −1)d}
Then,
n−1
X
k=0
a +kd =
n
2
(2a +(n −1)d) (10)
Geometric Series
Suppose we have the following geometric series,
{a + ar + ar2
+...+ ar(n−1)
}
Then,
n−1
X
k=0
ark
= a
µ
rn
−1
r −1
¶
(11)
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30.
Sequence and Series
Example
Solve the series
20
X
i=1
2i2
+7i
20
X
i=1
2i2
+7i =
20
X
i=1
2i2
+
20
X
i=1
7i
= 2
µ
20(20+1)(2(20)+1)
6
¶
+7
µ
20(20+1)
2
¶
= 2
µ
17220
6
¶
+7(210)
= 5740+1470
= 7210
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31.
Sequence and Series
Example
Find the sum of the geometric series: 2+8+32+128+...+8192
1 We know that
8
2
= 4 and that
32
8
= 4, so r = 4 for this geometric series.
2 The initial value represents our a value, so a = 2.
3 Before we can write the series in summation notation, we must determine the upper
limit of the summation.
8192 = arn−1
= (2)(4n−1
)
4096 = 4n−1
46
= 4n−1
6 = n −1
n = 7
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32.
Sequence and Series
So, we can rewrite the series as
P6
k=0
2(4k
). From the formula for the sum of a geometric
series,
6
X
k=0
2(4k
) = 2
µ
(4)7
−1
4−1
¶
= 2(5461)
= 10922
Therefore, the sum of the series, 2+8+32+128+...+8192 is 10922
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33.
Sequence and Series
Binomial Series
In this final section of this chapter we are going to look at another series representation
for a function. Before we do this let’s first recall the following theorem.
Binomial Theorem
If n is any positive integer then,
(a +b)n
=
n
X
i=0
Ã
n
i
!
an−i
bi
= an
+nan−1
b +
n (n −1)
2!
an−2
b2
+···+nabn−1
+bn
where, Ã
n
i
!
=
n (n −1)(n −2)···(n −i +1)
i!
; i = 1,2,3,...n
Ã
n
0
!
= 1
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34.
Sequence and Series
Example
(2x −3)4
=
4
X
i=0
Ã
4
i
!
(2x)4−i
(−3)i
=
Ã
4
0
!
(2x)4
+
Ã
4
1
!
(2x)3
(−3)+
Ã
4
2
!
(2x)2
(−3)2
+
Ã
4
3
!
(2x)(−3)3
+
Ã
4
4
!
(−3)4
= (2x)4
+4(2x)3
(−3)+
3
2
(2x)2
(−3)2
+4(2x)(−3)3
+(−3)4
= 16x4
−96x3
+216x2
−216x +81
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35.
Sequence and Series
Binomial Series
If k is any number and |x| < 1 then,
(1+ x)k
=
∞
X
n=0
Ã
k
n
!
xn
= 1+kx +
k (k −1)
2!
x2
+
k (k −1)(k −2)
3!
x3
+···
where, Ã
k
n
!
=
k (k −1)(k −2)···(k −n +1)
n!
; n = 1,2,3,...
Ã
k
0
!
= 1
Example
Write down the first four terms in the binomial series for
p
9− x
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36.
Sequence and Series
We have to rewrite the terms into the form (1+ xk)k
required, that is
p
9− x = 3
³
1−
x
9
´1
2
= 3
³
1+
³
−
x
9
´´1
2
So, k = 1
2 and xk = −(x/9)
Then the binomial series is given,
p
9− x = 3
³
1+
³
−
x
9
´´1
2
= 3
∞
X
n=0
Ã
1
2
n
!
³
−
x
9
´n
= 3
"
1+(
1
2
)(−
x
9
)+
1
2 (−1
2 )
2
(−
x
9
)
2
+
1
2 (−1
2 )(−3
2 )
6
(−
x
9
)
3
+···
#
So the first four terms are
= 3−
x
6
−
x2
216
−
x3
3888
−···
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37.
Sequence and Series
Exercise
1 Find the period of the following functions
1) f (x) = sin(2x), 2) f (x) = cos(−2x +π/3), 3) f (x) = x −sin(x).
2 Find the domain of:
1)f (x) =
1
2x −6
2)f (x) =
1− x
1+ x
3)f (x) =
x3
−2x
x(−x −6)
4)f (x) = 3x −1−
1
2x −6
5)f (x) =
x
1−2x + x2
6)f (x) =
x2
−2x
(x −3)(1− x2)
3 Determine whether the functions below are even, odd or neither.
1) f (x) = ex2
−1
+ln(|x|+1), 2) f (x) = x2
−2
x(1−x2)
, 3) f (x) = x2
sin(x)
4) f (x) = x
p
|x|−1, 5) f (x) = ln
¡
tanx −e|x|
¢
, 6) f (x) = x −1.
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38.
END OF LECTURE
THANK YOU
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