1. 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
2. Lecture Outline
1 Definitions
2 Types of Functions
Constant, Step, and Piecewise functions
Power functions
Polynomial Functions
Rational Functions
Algebraic Functions
Transcendental Functions
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3. Definitions
Outline of Presentation
1 Definitions
2 Types of Functions
Constant, Step, and Piecewise functions
Power functions
Polynomial Functions
Rational Functions
Algebraic Functions
Transcendental Functions
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4. 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
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5. 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
S R
s h
It could map a student to its height.
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6. 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
P N
p f (p)
It could map a product to its price.
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7. 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
P Name
c n = f (c)
It could map a country to its president.
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8. 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.
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9. 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.
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10. 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.
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11. 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.
Definition (Codomain)
Y, the set of all possible outputs, is called the codomain of f .
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14. 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)]
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15. Types of Functions
Outline of Presentation
1 Definitions
2 Types of Functions
Constant, Step, and Piecewise functions
Power functions
Polynomial Functions
Rational Functions
Algebraic Functions
Transcendental Functions
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16. 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.
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17. 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.
2 This function is defined for all real numbers: Its domain is Df = R.
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18. 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.
2 This function is defined for all real numbers: Its domain is Df = R.
3 Its range is {c}.
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19. 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.
2 This function is defined for all real numbers: Its domain is Df = R.
3 Its range is {c}.
4 Its codomain is R.
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20. Types of Functions Constant, Step, and Piecewise functions
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
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21. Types of Functions Constant, Step, and Piecewise functions
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.
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22. Types of Functions Constant, Step, and Piecewise functions
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.
1 Any number x is assigned to a unique real number: f is a function.
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23. Types of Functions Constant, Step, and Piecewise functions
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.
1 Any number x is assigned to a unique real number: f is a function.
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].
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24. Types of Functions Constant, Step, and Piecewise functions
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.
1 Any number x is assigned to a unique real number: f is a function.
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).
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25. Types of Functions Constant, Step, and Piecewise functions
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
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26. Types of Functions Constant, Step, and Piecewise functions
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.
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27. Types of Functions Constant, Step, and Piecewise functions
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.
1 g assigns two values to x0 = −4.
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28. Types of Functions Constant, Step, and Piecewise functions
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.
1 g assigns two values to x0 = −4.
2 That is g(−4) = {1,2}.
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29. Types of Functions Constant, Step, and Piecewise functions
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.
1 g assigns two values to x0 = −4.
2 That is g(−4) = {1,2}.
3 Thus, g is NOT a function.
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30. Types of Functions Constant, Step, and Piecewise functions
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.
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31. Types of Functions Constant, Step, and Piecewise functions
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.
2 Df = [0,+∞)∪(−∞,0) = (−∞,∞) = R.
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32. Types of Functions Constant, Step, and Piecewise functions
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.
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].
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33. Types of Functions Constant, Step, and Piecewise functions
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
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34. 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}).
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35. 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}).
Some examples of Power Functions
1 Monomial Functions are power function with positive power, that is p ∈ Z+
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36. 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}).
Some examples of Power Functions
1 Monomial Functions are power function with positive power, that is p ∈ Z+
2 Reciprocal Functions are power function with negative power, that is p ∈ Z−
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37. 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}).
Some examples of Power Functions
1 Monomial Functions are power function with positive power, that is p ∈ Z+
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
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38. Types of Functions Power functions
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,+∞)
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39. Types of Functions Power functions
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
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40. Types of Functions Power functions
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
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41. Types of Functions Power functions
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
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42. Types of Functions Power functions
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
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43. Types of Functions Power functions
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
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44. Types of Functions Power functions
Reciprocal functions: Graph of 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−2
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45. Types of Functions Power functions
Reciprocal functions: Graph of f (x) = x−3
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
Domain = R−{0}
Range = R−{0}
f (x) = x−3
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46. Types of Functions Power functions
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+
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47. Types of Functions Power functions
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
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48. Types of Functions Polynomial 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 .
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49. Types of Functions Polynomial 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 .
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.
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50. Types of Functions Polynomial Functions
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,+∞)
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51. Types of Functions Polynomial Functions
Polynomial Functions with 3 as highest power
−5 −4 −2 2 4
−3
−2
−1
1
2
3
0
g(x) = 1−2x −3x2
+2x3
Dg = R,
R(g) = R
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52. Types of Functions Polynomial Functions
Polynomial Functions with 4 as highest power
−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,+∞)
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55. 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.
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56. 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.
Example
1)f (x) =
p
5−2x 2)f (x) =
p
x −
1
x −1
3)f (x) =
2− x
p
x −1−2
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57. Types of Functions Algebraic Functions
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
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58. Types of Functions Transcendental Functions
Transcendental Functions
Definition
Transcendental functions are functions that are not algebraic.
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59. Types of Functions Transcendental Functions
Transcendental Functions
Definition
Transcendental functions are functions that are not algebraic.
Transcendental functions can be expressed in algebra in terms of an infinite sequence.
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60. Types of Functions Transcendental Functions
Transcendental Functions
Definition
Transcendental functions are functions that are not algebraic.
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
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61. Types of Functions Transcendental 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,+∞)
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62. Types of Functions Transcendental 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,+∞)
Example
1)
¡2
3
¢x
2) 2x
3) 3−x
4)
p
7
x
5) ex
6) e−x
.
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63. Types of Functions Transcendental 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,+∞)
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
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69. Types of Functions Transcendental Functions
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.
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70. Types of Functions Transcendental Functions
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.
Example
1) log2
3
x 2) log2 x 3) log1/3 x 4) logp
7 x 5) loge x 6) log1/e x.
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71. Types of Functions Transcendental Functions
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.
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)
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75. Types of Functions Transcendental Functions
Logarithmic Functions
−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
log2 x
log1/3 x
76. Types of Functions Transcendental Functions
Logarithmic Functions
−1 1 2 4 6 8 9
−3
−2
−1
1
2
3
0
log2 x
log1/3 x
log2
3
x
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77. Types of Functions Transcendental Functions
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:
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78. Types of Functions Transcendental Functions
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:
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)
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79. Types of Functions Transcendental Functions
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:
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
Dr. Gabby (KNUST-Maths) Functions 42 / 51
80. Types of Functions Transcendental Functions
Trigonometric Functions: sin and csc
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
sinx
81. Types of Functions Transcendental Functions
Trigonometric Functions: sin and csc
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
sinx
cscx
Dr. Gabby (KNUST-Maths) Functions 43 / 51
82. Types of Functions Transcendental Functions
Trigonometric Functions: cos and sec
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
cosx
83. Types of Functions Transcendental Functions
Trigonometric Functions: cos and sec
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
cosx
secx
Dr. Gabby (KNUST-Maths) Functions 44 / 51
84. Types of Functions Transcendental Functions
Trigonometric Functions: tan and cot
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
0
tanx
85. Types of Functions Transcendental Functions
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
Dr. Gabby (KNUST-Maths) Functions 45 / 51
87. Types of Functions Transcendental Functions
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)
Dr. Gabby (KNUST-Maths) Functions 47 / 51
88. Types of Functions Transcendental Functions
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
Dr. Gabby (KNUST-Maths) Functions 47 / 51
89. Types of Functions Transcendental Functions
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
Dr. Gabby (KNUST-Maths) Functions 48 / 51
90. Types of Functions Transcendental Functions
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
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
Dr. Gabby (KNUST-Maths) Functions 48 / 51
91. Types of Functions Transcendental Functions
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
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
Dr. Gabby (KNUST-Maths) Functions 48 / 51
92. Types of Functions Transcendental Functions
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
Dr. Gabby (KNUST-Maths) Functions 49 / 51
93. Types of Functions Transcendental Functions
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).
Dr. Gabby (KNUST-Maths) Functions 50 / 51