2. Functions I.pdf

FUNCTIONS I
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 / 51

Lecture Outline
1 Definitions
2 Types of Functions
Constant, Step, and Piecewise functions
Power functions
Polynomial Functions
Rational Functions
Algebraic Functions
Transcendental Functions

Definitions
Outline of Presentation
1 Definitions
Power functions
Rational Functions
Algebraic Functions

Definitions
Definition
1 Functions are mostly used to describe dependence between quantities.
2 In general, a function is a map between two sets that assigns to each element in the
first set a unique element in the second set.
input output
f
X Y
x y

Definitions
Definition
input output
f
S R
s h
It could map a student to its height.

Definitions
Definition
input output
f
P N
p f (p)
It could map a product to its price.

Definitions
Definition
input output
f
P Name
c n = f (c)
It could map a country to its president.

Definitions
Definition
Definition
1 A real-valued function f assigns a unique real number y to each input x.
2 If the function f is defined from a set X to Y, then we write
f : X → Y
x 7→ y = f (x)
The dependence could be described either by words, graphs, an equation or a tabulation.

Definitions
Definition
Definition
1 A real-valued function f assigns a unique real number y to each input x.
2 If the function f is defined from a set X to Y, then we write
f : X → Y
x 7→ y = f (x)
The dependence could be described either by words, graphs, an equation or a tabulation.
Remark
1 Uniqueness here means an input cannot yield more than one output i.e. x 7→ y1, y2 is
not allowed.
2 However, two different inputs x1 and x2 can be assigned to the same output y.

Definitions
Definition: If x → f → y
Definition (Domain)
The domain Df of a function f , is the set of all possible inputs where f is defined.

Definitions
Definition (Domain)
Definition (Codomain)
Y, the set of all possible outputs, is called the codomain of f .

Definitions
Definition (Domain)
Definition (Range)
The set of all assigned outputs, f (X ) =
©
f (x) | x ∈ X
ª
, is called the range/image of f .

Definitions
Definition (Domain)
Definition (Range)
The set of all assigned outputs, f (X ) =
©
f (x) | x ∈ X
ª
, is called the range/image of f .
Definition (Graph)
The graph of the function y = f (x) is a pictorial representation of the function. It is the
collection of the points (x, f (x)). It represents a curve in the Cartesian plane.

Definitions
Composition of Functions
1 A composite function is generally a function that is written inside another function.
2 Composition of a function is done by substituting one function into another function.
Example
1 f [g(x)] is the composite function of f (x) and g(x).
2 The composite function f [g(x)] is read as f of g of x.
3 The function g(x) is called an inner function and the function f (x) is called an outer
function.
4 f [g(x)] ̸= g[f (x)]

Types of Functions
Outline of Presentation
1 Definitions
Power functions
Rational Functions
Algebraic Functions

Types of Functions Constant, Step, and Piecewise functions
Constant functions
f : R → R
x 7→ c
1 Any real number x is assigned to the unique real number c: f is a function, but
specifically, f is a constant function.

Constant functions
f : R → R
x 7→ c
2 This function is defined for all real numbers: Its domain is Df = R.

Constant functions
f : R → R
x 7→ c
3 Its range is {c}.

Constant functions
f : R → R
x 7→ c
3 Its range is {c}.
4 Its codomain is R.

Constant function graph
Definition
A constant function is a function whose value is the same for every input value
f : R → R
x 7→ 1
−5 −4 −2 2 4
0

Step Function (or staircase function)
Definition
They are function that increases or decreases abruptly from one constant value to another.
f : R → (−∞,10)
x 7→ f (x) =







2 if x≤−4
1 if −4<x≤0
−1 if 0<x ≤ 2.

Definition
f : R → (−∞,10)
x 7→ f (x) =







2 if x≤−4
1 if −4<x≤0
−1 if 0<x ≤ 2.
1 Any number x is assigned to a unique real number: f is a function.

Definition
f : R → (−∞,10)
x 7→ f (x) =







2 if x≤−4
1 if −4<x≤0
−1 if 0<x ≤ 2.
2 This function is defined for all real numbers x ≤ −4 or −4 < x ≤ 0 or 0 < x ≤ 2. That is
Df = (−∞,−4]∪(−4,0](0,2] = (−∞,2].

Definition
f : R → (−∞,10)
x 7→ f (x) =







2 if x≤−4
1 if −4<x≤0
−1 if 0<x ≤ 2.
2 This function is defined for all real numbers x ≤ −4 or −4 < x ≤ 0 or 0 < x ≤ 2. That is
Df = (−∞,−4]∪(−4,0](0,2] = (−∞,2].
3 Its range is {−1,1,2} and its codomain is (−∞,10).

Step Function graph
f : R → (−∞,10)
x 7→ f (x) =







2 if x≤−4
1 if −4<x≤0
−1 if 0<x ≤ 2.
−5 −4 −2 2 4
−2
2
3
0

Example of Non Step Function
Consider the following relation
g : R → R
x 7→ g(x) =







2 if x≤−4
1 if −4≤x≤0
−1 if 0<x ≤ 2.

g : R → R
x 7→ g(x) =







2 if x≤−4
1 if −4≤x≤0
−1 if 0<x ≤ 2.
1 g assigns two values to x0 = −4.

g : R → R
x 7→ g(x) =







2 if x≤−4
1 if −4≤x≤0
−1 if 0<x ≤ 2.
2 That is g(−4) = {1,2}.

g : R → R
x 7→ g(x) =







2 if x≤−4
1 if −4≤x≤0
−1 if 0<x ≤ 2.
2 That is g(−4) = {1,2}.
3 Thus, g is NOT a function.

Piecewise functions
Definition
Piecewise functions are defined by different functions for different intervals of the domain.
f : R → R
x 7→ f (x) =
(
−x +2 if x ≥ 0
2x +2 if x < 0.
1 f is a piecewise function.

Piecewise functions
Definition
f : R → R
x 7→ f (x) =
(
−x +2 if x ≥ 0
2x +2 if x < 0.
2 Df = [0,+∞)∪(−∞,0) = (−∞,∞) = R.

Piecewise functions
Definition
f : R → R
x 7→ f (x) =
(
−x +2 if x ≥ 0
2x +2 if x < 0.
2 Df = [0,+∞)∪(−∞,0) = (−∞,∞) = R.
3 Its range is I = { −x +2 | x ≥ 0}∪{ 2x +2 | x < 0}.
x ≥ 0 =⇒ −x ≤ 0 =⇒ −x +2 ≤ 2.
x < 0 =⇒ 2x +2 < 2.
Therefore, I = (−∞,2]∪(−∞,2) = (−∞,2].

Piecewise functions graph
f : R → R
x 7→ f (x) =
(
−x +2 if x ≥ 0
2x +2 if x < 0.
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0

Types of Functions Power functions
Power Functions
Definition
Power functions are functions of the form of
f (x) = axp
(1)
where p is any real number (p ∈ R) and a is a non-zero real number, that is (a ∈ R−{0}).

Power Functions
Definition
f (x) = axp
(1)
Some examples of Power Functions
1 Monomial Functions are power function with positive power, that is p ∈ Z+

Power Functions
Definition
f (x) = axp
(1)
2 Reciprocal Functions are power function with negative power, that is p ∈ Z−

Power Functions
Definition
f (x) = axp
(1)
2 Reciprocal Functions are power function with negative power, that is p ∈ Z−
3 Radical Functions are power functions where the degree p is of the form 1
n and n ∈ N

Power Functions
Example
f (x) Domain Range f (x) Domain Range
x2n
R R+
2n
p
x R+ R+
x2n+1
R R 2n+1
p
x R R
1
x2n
R−{0} R+ −{0}
1
2n
p
x
R+ −{0} R+ −{0}
1
x2n+1
R−{0} R−{0}
1
2n+1
p
x
R−{0} R−{0}
n ∈ Z+, R−{0} = (−∞,0)∪(0,+∞), R+ = [0,+∞), R+ −{0} = (0,+∞)

Monomials: Graph of f (x) = 1
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R
Range = {1}
f (x) = 1

Monomials: Graph of f (x) = x
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R
Range = R f (x) = x

Monomials: Graph of f (x) = x2
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R
Range = R+
f (x) = x2

Monomials: Graph of f (x) = x3
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R
Range = R
f (x) = x3

Reciprocal functions: Graph of f (x) = x−1
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R−{0}
Range = R−{0}
f (x) = x−1

−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R−{0}
Range = R+ −{0}
f (x) = x−2

−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R−{0}
Range = R−{0}
f (x) = x−3

Radical functions: Graph of f (x) =
p
x = x1/2
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
f (x) =
p
x
Domain = R+
Range = R+

Radical functions: Graph of f (x) = 3
p
x = x1/3
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
g(x) =
3
p
x
Domain = R
Range = R

Types of Functions Polynomial Functions
Definition
Sum of monomials of different degrees is called a polynomial. If f is a polynomial, then
f (x) = a0 + a1x + a2x2
+···+ anxn
(2)
1 n is a non-negative integer called the degree;
2 an is a non-zero real number;
3 ai ’s are called the coefficients of the polynomial f .

Definition
Sum of monomials of different degrees is called a polynomial. If f is a polynomial, then
f (x) = a0 + a1x + a2x2
+···+ anxn
(2)
1 n is a non-negative integer called the degree;
2 an is a non-zero real number;
3 ai ’s are called the coefficients of the polynomial f .
Note
1 The domain of a polynomial function is R.
2 Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic functions
respectively.

Polynomial Functions with 2 as highest power
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
f (x) = 1/2+ x2
,
Df = R,
R(f ) = [1/2,+∞)

−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
g(x) = 1−2x −3x2
+2x3
Dg = R,
R(g) = R

−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
h(x) = (x −1)(−x +3)2
(x),
Dh = R,
R(h) = [−1.6,+∞)

Types of Functions Rational Functions
Rational Functions
Definition
1 A rational function is a ratio f (x) = P(x)
Q(x) , where P and Q are polynomials.
2 The domain of f is Df =
©
x ∈ R | Q(x) ̸= 0
ª
.
Example
1 f (x) =
3x
x3 −1
2 f (x) =
3x3
− x6
2x2 − x +2

Types of Functions Rational Functions
Rational Functions
−5 −4 −3 −2 −1 1 3
2 4
−3
−2
−1
1
2
3
0
f (x) = x3
−2x
x2−1
,
Df = R−{−1,1},
R(f ) = R

Types of Functions Algebraic Functions
Algebraic functions
Definition
1 An algebraic function is a function that can be defined as the root of a polynomial
equation.
2 An algebraic function is constructed by taking sums, products, and quotient of
polynomials.

Algebraic functions
Definition
1 An algebraic function is a function that can be defined as the root of a polynomial
equation.
2 An algebraic function is constructed by taking sums, products, and quotient of
polynomials.
Example
1)f (x) =
p
5−2x 2)f (x) =
p
x −
1
x −1
3)f (x) =
2− x
p
x −1−2

Algebraic functions
−5 −4 −3 −2 −1 1 3
2 4
−3
−2
−1
1
2
3
0
f (x) =
p
x − 1
x−1 ,
Df = R+ −{1},
R(f ) = R

Types of Functions Transcendental Functions
Definition
Transcendental functions are functions that are not algebraic.

Definition
Transcendental functions can be expressed in algebra in terms of an infinite sequence.

Definition
Transcendental functions can be expressed in algebra in terms of an infinite sequence.
Example
1 Exponential functions
2 Logarithmic functions
3 Trigonometric functions
4 Hyperbolic functions
5 Inverse of these functions

Exponential Functions
Definition
The function f (x) = ax
, where a > 0 and a ̸= 1, is called exponential function with base a.
The domain of an exponential function is R and the range is (0,+∞)

Definition
Example
1)
¡2
3
¢x
2) 2x
3) 3−x
4)
p
7
x
5) ex
6) e−x
.

Definition
Example
1)
¡2
3
¢x
2) 2x
3) 3−x
4)
p
7
x
5) ex
6) e−x
.
Note
e is mathematical constant called the Euler number approximated as 2.71828

−5 −4 −3 −2 −1 1 3
2 4
−1
1
2
3
4
5
0
ex

−5 −4 −3 −2 −1 1 3
2 4
−1
1
2
3
4
5
0
ex
e−x

−5 −4 −3 −2 −1 1 3
2 4
−1
1
2
3
4
5
0
2x

−5 −4 −3 −2 −1 1 3
2 4
−1
1
2
3
4
5
0
2x
3−x

−5 −4 −3 −2 −1 1 3
2 4
−1
1
2
3
4
5
0
2x
3−x
¡2
3
¢x

Logarithmic Functions
Definition
The function f (x) = loga(x), where a > 0 and a ̸= 1, is called logarithmic function with base
a.
The domain of a logarithmic function is (0,+∞) and the range is R.

Definition
a.
Example
1) log2
3
x 2) log2 x 3) log1/3 x 4) logp
7 x 5) loge x 6) log1/e x.

Definition
a.
Example
1) log2
3
x 2) log2 x 3) log1/3 x 4) logp
7 x 5) loge x 6) log1/e x.
Natural log
This is the log to the base e and it also called ln. That is
loge = ln (3)

−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
loge x

−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
loge x
log1/e x

−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
log2 x

−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
log2 x
log1/3 x

−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
log2 x
log1/3 x
log2
3
x

Trigonometric Functions
Definition
Trigonometric functions are also known as Circular Functions are functions of an angle of
a triangle. It means that the relationship between the angles and sides of a triangle are
given by these trigonometric functions. If x is an acute angle in a right triangle, then:

Definition
Some Basic Trig function:
1 sin(x) =
opposite
hypotenuse
2 cos(x) =
ad j acent
hypotenuse
3 tan(x) =
opposite
ad j acent
4 csc(x) = 1
sin(x)
5 sec(x) = 1
cos(x)
6 cot(x) = 1
tan(x)

Definition
Some Basic Trig function:
1 sin(x) =
opposite
hypotenuse
2 cos(x) =
ad j acent
hypotenuse
3 tan(x) =
opposite
ad j acent
4 csc(x) = 1
sin(x)
5 sec(x) = 1
cos(x)
6 cot(x) = 1
tan(x)
Some Trig Identities
1 cos2
x +sin2
x = 1
2 sec2
x −tan2
x = 1
3 csc2
x −cot2
x = 1
4 sin
¡
x ± y
¢
= sinx cos y ±cosx sin y
5 cos
¡
x ± y
¢
= cosx cos y ∓sinx sin y
6 tan
¡
x + y
¢
=
tanx +tan y
1+tanx tan y

Trigonometric Functions: sin and csc
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
sinx

Trigonometric Functions: sin and csc
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
sinx
cscx

Trigonometric Functions: cos and sec
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
cosx

Trigonometric Functions: cos and sec
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
cosx
secx

Trigonometric Functions: tan and cot
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
tanx

Trigonometric Functions: tan and cot
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
tanx
cotx
cotx

Remarks
f (x) Domain Range
sin R [−1,1]
cos R [−1,1]
tan R−
©
...,−3π
2 ,−π
2 , π
2 , 3π
2 ,...
ª
R
sec R−
©
...,−3π
2 ,−π
2 , π
2 , 3π
2 ,...
ª
(−∞,−1)∪(1,+∞)
csc R−{...,−2π,−π,0,π,2π,...} (−∞,−1)∪(1,+∞)
cot R−{...,−2π,−π,0,π,2π,...} R
1 Dtan = Dsec = R−{ π/2+kπ | k ∈ Z}
2 Dcot = Dcsc = R−{ kπ | k ∈ Z}.

Transcendental: Inverse Trigonometric Functions
There are the functions
1 f (x) = sin−1
x (also called arc sine or arcsin)
2 f (x) = cos−1
x (arc cosine or arccos)
3 f (x) = tan−1
x (arc tangent or arctan)
4 f (x) = csc−1
x (arc cosec)
5 f (x) = sec−1
x (arc secant)
6 f (x) = cot−1
x (arc cotangent)

Transcendental: Inverse Trigonometric Functions
There are the functions
1 f (x) = sin−1
x (also called arc sine or arcsin)
2 f (x) = cos−1
x (arc cosine or arccos)
3 f (x) = tan−1
x (arc tangent or arctan)
4 f (x) = csc−1
x (arc cosec)
5 f (x) = sec−1
x (arc secant)
6 f (x) = cot−1
x (arc cotangent)
Note
1 y = sin−1
x ⇔ x = sin y
2 y = cos−1
x ⇔ x = cos y

Transcendental: Hyperbolic and Inverse Hyperbolic Functions
These are functions defined in terms of the exponential functions
Hyperbolic
1 sinhx =
ex
−e−x
2
2 coshx =
ex
+e−x
2
3 tanhx =
sinhx
coshx
4 csch x =
1
sinhx
5 sech x =
1
cosh
6 cothx =
1
tanh

Hyperbolic
1 sinhx =
ex
−e−x
2
2 coshx =
ex
+e−x
2
3 tanhx =
sinhx
coshx
4 csch x =
1
sinhx
5 sech x =
1
cosh
6 cothx =
1
tanh
Inverse Hyperbolic
1 sinh−1
x
2 cosh−1
x
3 tanh−1
x
4 csch−1
x
5 sech−1
x
6 coth−1
x

Hyperbolic
1 sinhx =
ex
−e−x
2
2 coshx =
ex
+e−x
2
3 tanhx =
sinhx
coshx
4 csch x =
1
sinhx
5 sech x =
1
cosh
6 cothx =
1
tanh
Inverse Hyperbolic
1 sinh−1
x
2 cosh−1
x
3 tanh−1
x
4 csch−1
x
5 sech−1
x
6 coth−1
x
Some identities
1 cosh2
x −sinh2
x = 1
2 tanh2
x +sech2
x = 1

Hyperbolic Functions
Identities
1 cosh2
x −sinh2
x = 1
2 tanh2
x +sech2
x = 1
3 sinh
¡
x + y
¢
= sinhx cosh y +coshx sinh y
4 cosh
¡
x + y
¢
= coshx cosh y +sinhx sinh y

Exercise
1 Which of the following are not polynomial functions?
a. f (x) = 1 b. f (x) = x2
+ x−1
+1
c. f (x) = −2x3
+ x1/2
−1 d. f (x) = x4
p
5−π.
2 Find the range of the following polynomial functions:
a. f (x) = x2
+6 b. f (x) = −2x4
−6
c. f (x) = −2x3
+1 d. f (x) =
¯
¯−2x3
+1
¯
¯
e. f (x) = 3−4x,Df = (−2,8]
f . f (x) = (2x −1)2
+1,Df = (−∞,−1)∪(1,+∞).
3 Find the domain of:
1) f (x) = lnx, 2) f (x) = log5(1−3x),
3) f (x) = e
1
x+1
−x
, 4) f (x) = ex2
−1
+ln(|x|+1).

END OF LECTURE
THANK YOU

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