Algebraic and Transcendental Functions
EEE102: Research 01
Abdul Aziz Guevarra Mabaning
Donald Billy Cabriana Abugan II
IEEE Member
Department of Electrical Engineering and Technology
Department of Electrical Engineering and Technology
Mindanano State University - Iligan Institute of Technology
Andres Bonifacio Ave, Iligan City, 9200 Lanao del Norte Mindanano State University - Iligan Institute of Technology
Andres Bonifacio Ave, Iligan City, 9200 Lanao del Norte
donaldbillyii.abugan@g.msuiit.edu.ph
abdulazis.mabaning@g.msuiit.edu.ph
Abstract—In this paper, we will discuss two kinds of mathematical functions: algebraic and transcendental. We will explain what
algebraic and transcendental functions are and show how to solve
their equations. In the end of this paper, we will already have
a basic understanding of what are algebraic and transcendental
functions and their solutions.
•
I. A LGEBRAIC F UNCTIONS
•
A. Definition
An algebraic function is a function in mathematics that can
be defined as the root of a polynomial equation. Algebraic
functions only use algebraic operations like addition, subtraction, multiplication, and division, as well as fractional or
rational exponents. You can consider it like a machine that
accepts real numbers, performs mathematical operations, and
outputs other numbers.
B. Types of Algebraic Functions and Their Solutions
a) Polynomial Functions: Polynomial functions are a
type of algebraic function with a polynomial definition. Examples of polynomial functions are: Linear functions, quadratic
functions, cubic functions, biquadratic functions, quintic functions and etc. Examples of polynomial functions are:
f (x) = 4x–2
(Linearf unction)
f (x) = 2x2 + 3x + 1
3x3 − 6x2 + 4x–9
(QuadraticF unction)
(CubicF unction)
•
b) Rational Functions: Rational functions are a type of
algebraic function that can be expressed as the quotient or
division of two polynomial functions. In other words, it’s a
fraction where both the numerator and the denominator are
polynomial expressions. They have the general form: f(x) =
p(x)/q(x), where p(x) and p(x) are polynomials in x. Examples
of rational functions are:
f (x) =
(2)
•
Linear Polynomial: For a linear equation in the form of
ax + b = 0, solve for x by isolating it.
Quadratic Polynomial: For a quadratic equation in the
form of ax2 + bx + c = 0,
(5)
2x2 + 3x − 5
x2 − 6x + 9
(6)
3x2 + 5
2x − 1
(7)
f (x) =
(3)
Solving rational functions typically involves finding the
values of the variable (usually denoted as ”x”) that make the
rational function equal to zero or undefined.
•
•
•
2x + 3
x+4
f (x) =
(1)
There are many methods for solving polynomial functions.
The goal for solving polynomial functions is to find the
values of the variable (usually denoted as x) that make the
equation equal to 0. These values are called the ”roots”
or ”solutions” of the polynomial equation. Generally, the
approach to solving polynomial functions depends on the
degree of the polynomial:
you can use the quadratic formula:
√
−b ± b2 − 4ac
x=
(4)
2a
Factoring: For polynomial equations that can be factored, you can factor the polynomial and set each factor
equal to zero to find the roots.
Synthetic Division: For polynomials with rational roots,
you can use synthetic division to test possible rational
roots and narrow down the search for solutions.
Graphical Approach Use graphing techniques to visualize where the polynomial intersects the x-axis (where
it equals zero).
Simplify the Function: Simplify the rational function by
canceling common factors in the numerator and denominator if possible.
Set the Denominator Equal to Zero: Determine the
values of ”x” that make the denominator equal to zero.
These values are called the ”restrictions” or ”excluded
values.” To do this, solve the equation obtained by setting
the denominator equal to zero. These values must be
•
•
•
excluded from the domain because division by zero is
undefined.
Find the Zeros of the Numerator: Determine the values
of ”x” that make the numerator equal to zero. These
values are called the ”zeros” or ”roots” of the rational
function.
Analyze the Critical Points: Critical points are the
values of ”x” where the function may have vertical
asymptotes, holes, or horizontal asymptotes. These points
are usually where the zeros of the denominator or numerator occur.
Find the Domain Determine the domain of the rational
function. The domain is all real numbers except for the
excluded values.
c) Power Functions: Power functions are mathematical
expressions that show how a value changes when it’s raised
to a constant exponent. They have the form:
f (x) = kxa
(8)
where k and a can be any real number. Because ’a’ is a real
number, the exponent might be either an integer or a rational
number. Examples of power functions are:
f (x) = 2x2
f (x) = 2x−1 =
√
•
•
B. Types of Transcendental Functions and Their Solutions
a) Exponential Functions: An exponential function is a
mathematical function with the formula f (x) = ax , where ”x”
is a variable and ”a” is a constant that is called the function’s
base and must be greater than zero. The transcendental number
e, which is approximately equal to 2.71828, is the most
often used exponential function basis. Examples of exponential
functions are:
f (x) = 2x
1
f (x) = x = 2−x
2
f (x) = 2x+3
(9)
2
x
x − 4 = (x − 4)1/2
(10)
(11)
Solving power functions typically involves finding the values
of the variable (usually denoted as ”x”) that make the
function equal to a specific value or satisfy certain conditions.
The approach to solving power functions depends on the
specific equation or problem you’re working on. Here are
some common techniques and steps to solve power functions:
•
II. T RANSCENDENTAL F UNCTIONS
A. Definition
When a function cannot be expressed in terms of a
finite combination of algebraic operations such as addition,
subtraction, division, or multiplication raising to a power and
extracting a root, it is said to be transcendental. Transcendental
functions include log x, sin x, cos x, and others. Exponential
functions, logarithmic functions, trigonometric functions,
hyperbolic functions, and the inverse of all of these functions
are the most well-known transcendental function examples.
Gamma, Elliptic, and Zeta functions are less well-known
transcendental function examples.
Equating to a Value (e.g., finding roots):
For Positive Integer Exponents (n > 0): To solve
equations like (axn = b) where a, b, and n are
constants, take the nth root of both sides (if n is even,
consider both positive and negative roots).
For Negative Integer Exponents (n < 0): Inverse
the equation and take the nth root of both sides.
Equating to 0: Set the power function equal to zero and
solve for x.
Inequalities: When solving inequalities involving power
functions, treat them similarly to equations, but consider
the sign of the function for different intervals
(12)
(13)
(14)
To solve exponential functions, equations will typically
have the variable in the exponent. The specific approach
to solving these equations depends on the form of the
exponential equation. Here are some common types of
exponential equations and methods to solve them:
•
•
•
•
Basic Exponential Equations: Basic exponential equations are those where the variable is in the exponent
and the base is a constant. To solve, you can take the
logarithm (usually natural logarithm) of both sides. For
example, if you have 2x = 8, taking the natural logarithm
of both sides yields x ln(2) = ln(8). Then, solve for x.
Exponential Equations with Different Bases: When
the bases of the exponents are different, you can use
logarithmic properties to solve the equation. For example,
to solve 3x = 5x , you can take the natural logarithm of
both sides to get x ln(3) = x ln(5), and then solve for x.
Exponential Equations with Exponential Terms: Some
equations involve exponential terms on both sides, like
ex = 32x . In such cases, you might take the natural
logarithm of both sides to simplify and solve.
Exponential Inequalities: For inequalities involving exponential functions, you can often use similar techniques
as for equations, but you need to consider the sign of the
inequality when taking logarithms. For example, to solve
2x > 16, you take the natural logarithm of both sides
but remember to reverse the inequality sign when 2x is
negativ
b) Logarithmic Functions: The logarithmic function
is the inverse function of exponentiation in mathematics. As
defined by the logarithmic function. For x > 0, a > 0, and
a = 1, y = loga (x) if and only if x = ay . Then the function
is given by f (x) = loga (x)
The base of the logarithm is a. This can be read as log
base a of x. The most 2 common bases used in logarithmic
functions are base 10 and base e. The logarithmic function to
the base e is called the natural logarithmic function and it is
denoted by loge or ln. f (x) = loge (x) = ln(x). Examples of
Logarithmic Functions are:
f (x) = log10 (x)
f (x) = ln(x)
f (x) = log2 (x)
(CommonLogarithm)
(N aturalLogarithm)
(15)
(16)
1) Algebraic Trigonometric Equations:
These equations involve trigonometric functions and algebraic expressions. The key is to isolate the trigonometric function on one side of the equation and use inverse
trigonometric functions or trigonometric identities to
find solutions.
2) Quadratic Trigonometric Equations:
Quadratic trigonometric equations involve trigonometric
functions and quadratic expressions. You can use factoring, the quadratic formula, or trigonometric identities to
solve them.
3) Trigonometric Identities:
Trigonometric identities can simplify trigonometric
equations. Use identities like the double-angle, halfangle, or Pythagorean identities to simplify and solve
equations.
(GeneralLogarithmicF unction) (17)
R EFERENCES
To solve logarithmic equations, you typically follow these
steps:
1) Isolate the Logarithm:
Start by isolating the logarithmic expression on one
side of the equation. For example, consider the equation
logb (x) = c, where b is the base, and c is a known
constant.
2) Convert to Exponential Form:
To eliminate the logarithm, rewrite the equation in exponential form. For a logarithmic equation logb (x) = c,
the exponential form is x = bc .
This step essentially ”undoes” the logarithm, revealing
the value of the variable.
3) Calculate the Solution: Calculate the value of bc to find
the solution for x.
For example, if you have log2 (x) = 3, you can convert
it to x = 23 and calculate x = 8.
c) Trigonometric Functions: Trigonometric functions,
often known as circular functions, are simple functions of a
triangle’s angle. These trig functions define the relationship
between the angles and sides of a triangle. Sine, cosine,
tangent, cotangent, secant, and cosecant are the fundamental
trigonometric functions.
Formulas for Angle θ Reciprocal Identities
Side
sin θ = Opposite
sin θ = csc1 θ
Hypotenuse
Adjacent Side
cos θ = Hypotenuse
cos θ = sec1 θ
Side
tan θ = Opposite
tan θ = cot1 θ
Adjacent
Adjacent Side
1
cot θ = Opposite
cot θ = tan
θ
Hypotenuse
1
sec θ = Adjacent Side
sec θ = cos θ
csc θ = Hypotenuse
csc θ = sin1 θ
Opposite
Trigonometric equations are classified into several types,
and the solution may vary depending on the type of equation.
Here are some examples and methods for solving trigonometric equations:
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