What are Transcendental Functions?
In mathematics, when a function is not expressible in terms of a finite combination of algebraic operation of addition, subtraction, division, or multiplication raising to a power and extracting a root, then they are said to be transcendental functions. Some of the examples of transcendental functions can be log x, sin x, cos x, etc. These functions that are non-algebraic in nature can only be expressed in terms of infinite series.
In Mathematics, transcendental functions are the analytical functions that are not algebraic, and hence do not satisfy the polynomial equation. In other words, transcendental functions cannot be expressed in terms of finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting the roots. The functions such as logarithmic, trigonometric functions, and exponential functions are a few examples of transcendental functions.
The transcendental functions can be expressed in algebra only in the terms of an infinite sequence. Hence, the term transcendental means non-algebraic.
Define Transcendental Functions
The transcendental function can be defined as a function that is not algebraic and cannot be expressed in terms of a finite sequence of algebraic operations such as sin x.
The most familiar transcendental functions examples are the exponential functions, logarithmic functions, trigonometric functions, hyperbolic functions, and inverse of all these functions. The less familiar transcendental functions examples are Gamma, Elliptic, and Zeta functions.
What is a Transcendental Equation?
A polynomial equation is an equation in the form of
\[x^{4} - 4x^{2} - 3 = 0, 4x^{2} - 3x + 9 = 0, and 2x^{3}- 5x^{2} - 7x + 3\] are some of the algebraic equations.
An equation containing polynomials, logarithmic functions, trigonometric functions, and exponential functions is known as transcendental equation.
\[tan x - e^{x} = 0, sin x - xe^{2x} = 0 , and xe^{x} = cos x\] are some of the transcendental equations examples.
Define Transcendental Equations
A transcendental equation is an equation into which transcendental functions (such as exponential, logarithmic, trigonometric, or inverse trigonometric) of one of the variables (s) have been solved for. Transcendental equations do not have closed-form solutions.
Transcendental equations examples includes: \[x =e^{-x}, x = cos x, 2^{x} = x^{2}\].
Transcendental Functions Examples With Solutions
1. Find dy/dx for the function y = In(tan x + sec x)
Solution:
dy/dx = x² (1/4x. 4) + In (4x). 2x
= x + 2x In ( 4x)
= x( 1 + 2 In (4x))
2. Calculate \[\lim_{x\rightarrow 0}\frac{secx-1}{sinx}\]
Solution:
As both numerator and denominator approaches to 0. Hence, applying L’s hospital rule, we get:
\[\lim_{^{2}x\rightarrow 0}\frac{secx-1}{sinx}\]
\[=\lim_{x\rightarrow 0}\frac{secx-tanx}{cosx}\]
\[=\lim_{x\rightarrow 0}\frac{1.0}{1}\]
= 1
FAQs on Transcendental Function
1. What is a transcendental function in mathematics?
A transcendental function is a type of function that cannot be expressed using a finite number of algebraic operations (addition, subtraction, multiplication, division, raising to a power, and taking a root). In simpler terms, it 'transcends' or goes beyond basic algebra. These functions are not solutions to any polynomial equation whose coefficients are themselves polynomials. This is in direct contrast to algebraic functions, like f(x) = x² + 3x - 2.
2. What are the main types of transcendental functions with examples?
The most common transcendental functions studied in the CBSE syllabus include several key categories. The main types and their examples are:
- Exponential Functions: Functions where the variable is the exponent, such as f(x) = eˣ or f(x) = 10ˣ.
- Logarithmic Functions: The inverse of exponential functions, like f(x) = log(x) or f(x) = ln(x).
- Trigonometric Functions: These relate an angle of a right-angled triangle to ratios of two side lengths. Examples include f(x) = sin(x), f(x) = cos(x), and f(x) = tan(x).
- Inverse Trigonometric Functions: The inverse functions of the trigonometric functions, such as f(x) = arcsin(x) or f(x) = tan⁻¹(x).
- Hyperbolic Functions: Analogues of trigonometric functions defined using the hyperbola, such as f(x) = sinh(x) and f(x) = cosh(x).
3. How do transcendental functions fundamentally differ from algebraic functions?
The fundamental difference lies in their definition and origin. Algebraic functions can always be described as the root of a polynomial equation. For example, the function y = √x + 1 is algebraic because it can be rewritten as the polynomial equation (y-1)² - x = 0. In contrast, transcendental functions like y = sin(x) or y = eˣ cannot be expressed this way. Their behaviour, such as the infinite periodicity of sin(x) or the rapid growth of eˣ, cannot be captured by finite polynomial expressions.
4. Why are transcendental functions so important in calculus?
Transcendental functions are crucial in calculus because they describe relationships and rates of change that algebraic functions cannot. Their derivatives and integrals have unique and fundamental properties. For instance:
- The function f(x) = eˣ is its own derivative, making it the cornerstone for solving differential equations that model natural growth and decay.
- The integral of f(x) = 1/x is the logarithmic function ln|x|, which provides a way to calculate areas under curves where algebraic methods fail.
- Trigonometric functions are essential for modelling any periodic or wavelike phenomena, and their calculus is fundamental to physics and engineering.
5. Where can we see examples of transcendental functions in the real world?
Transcendental functions model many phenomena in science, finance, and nature. For example:
- Exponential Growth/Decay (eˣ): Used to model compound interest, population growth, and radioactive decay.
- Logarithmic Scales (log x): The Richter scale for earthquakes, the pH scale for acidity, and the decibel scale for sound intensity are all logarithmic.
- Wave Phenomena (sin x, cos x): Used to describe sound waves, light waves, AC electricity circuits, and the oscillating motion of a pendulum.
6. Is a function involving a root, like y = √x, a transcendental function?
No, a function involving a root (a rational exponent) like y = √x is an algebraic function, not a transcendental one. This is a common point of confusion. Although it isn't a simple polynomial, it can be defined as the solution to a polynomial equation. In this case, by squaring both sides, we get y² = x, which can be written as y² - x = 0. Since it satisfies a polynomial relationship between x and y, it is classified as algebraic.
7. What makes the graph of a transcendental function different from a polynomial's graph?
The graphs of transcendental functions often exhibit behaviours not seen in polynomials. For example, a polynomial of degree 'n' can have at most 'n' roots and 'n-1' turning points. In contrast:
- Periodic functions like y = sin(x) have an infinite number of roots and turning points.
- Exponential functions like y = eˣ grow faster than any polynomial for large x and have a horizontal asymptote, a line the graph approaches but never touches.
- Logarithmic functions like y = ln(x) have a vertical asymptote and are only defined for positive x-values.
These features like periodicity and asymptotes are characteristic of many transcendental functions.
