Transcendental Function - Explanation, Equation and Examples

Transcendental function is a category of functions that cannot be expressed as finite polynomials, algebraic functions, or solutions of algebraic equations with rational coefficients. They are crucial in advanced mathematics and have applications across various scientific fields. These are non-algebraic functions.

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This Story also Contains

1. Transcendental Functions
2. Properties of Transcendental Functions
3. Applications of Transcendental Functions
4. Solved Examples Based On the Transcendental Functions

In this article, we will cover the concepts of the transcendental function especially exponential and logarithmic functions. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seven questions have been asked on this concept, including two in 2019, and five in 2023.

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.

A polynomial equation is an equation in the form of

1. $x^4-4 x^2-3=0$
2. $4 x^2-3 x+9=0$
3. $2 x^3-5 x^2-7 x+3=0$

are some of the algebraic equations.

An equation containing polynomials, logarithmic functions, trigonometric functions, and exponential functions is known as transcendental equation.

1. $\tan x-e^x=0$
2. $\sin x-x e^{2 x}=0$
3. $x e^x=\cos x$

are some of the transcendental equations examples.

Transcendental Functions Definition

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:

1. $x=e^{-x}$
2. $x=\cos x$

Types of Transcendental function

Exponential Function: function $f(x)$ such that $f(x)=a^x$ is known as an exponential function.
base: $a>0, a \neq 1$
domain : $x \in \mathbb{R}$
range : $\mathrm{f}(\mathrm{x})>0$

Property : If $a^x=a^y$,then $x=y$

Properties

• Continuity and Differentiability: It is defined for all x and is also both continuous and differentiable It has the property that its derivative is itself: (d/dx) e^x = e^x
• Growth Rate: Exponential functions grow exponentially fast, which means e^x increases much more rapidly than any polynomial as x becomes large.
• Series Representation: The exponential function can be expressed as an infinite series: e^x = \sum_{n=0}^{\infty} (xⁿ/ n!)

Logarithmic function: function $f(x)$ such that $f(x)=\log _a(x)$ is called logarithmic function
base: $a>0, a \neq 1$
domain : $\mathrm{x}>0$
range $: f(x) \in \mathbb{R}$

Properties of Logarithmic Function

1. $\log _e(a b)=\log _e a+\log _e b$
2. $\log _e\left(\frac{a}{b}\right)=\log _e a-\log _e b$
3. $\log _e a^{\mathrm{m}}=m \log _e a$
4. $\log _a a=1$
5. $\log _{\mathrm{b}^{\mathrm{m}}} \mathrm{a}=\frac{1}{\mathrm{~m}} \log _{\mathrm{b}} \mathrm{a}$
6. $\log _b a=\frac{1}{\log _{\mathrm{a}} \mathrm{b}}$
7. $\log _b a=\frac{\log _m a}{\log _m b}$
8. $\mathrm{a}^{\log _a m}=\mathrm{m}$
9. $\mathrm{a}^{\log _c \mathrm{~b}}=\mathrm{b}^{\log _c \mathrm{a}}$
10. $\log _{\mathrm{m}} \mathrm{a}=\mathrm{b} \Rightarrow \mathrm{a}=\mathrm{m}^{\mathrm{b}}$

• Inverse Relationship: The natural logarithm ln(x) satisfies e^ln(x) = x for x > 0.
• Derivative: Thus, the derivative of ln(x) = 1/x.
• Growth Rate: Functions of logarithmic type increase slower to some extent than all polynomial functions.

Trigonometric Functions

Trigonometry functions or circular functions are sine, cosine and tangent they are used in the analysis of periodic behavior. These functions also show time periodicity and are used nearly in all branches of science and application like physics, engineering, and signal processing.

Properties

• Periodicity: Functions such as sin(x) and cos(x) are periodic with a period of 2π.
• Differentiability: The derivatives of trigonometric functions follow specific patterns: [(d/dx) sin x] = cos x and [(d/dx) cos x] = -sinx

Examples

• Sine function: sin(x)
• Cosine function: cos(x)
• Tangent function: tan(x)

Hyperbolic Functions

Such functions, like hyperbolic sine sinh(x) and hyperbolic cosine cosh(x), are analogues of trigonometric functions but for a hyperbola.

Properties

• Analogous to Trigonometric Functions: Comparable to trigonometric functions’ relation to the unit circle, we have hyperbolic functions in relation to the hyperbola.
• Derivatives: The derivatives of hyperbolic functions are similar to their trigonometric counterparts:
  1. $\frac{d}{d x} \sinh x=\cosh x$
  2. $\frac{d}{d x} \cosh x=\sinh x$

Examples

• Hyperbolic sine: $\sinh x=\frac{e^x-e^{-x}}{2}$
• Hyperbolic cosine: $\cosh x=\frac{e^x+e^{-x}}{2}$

Properties of Transcendental Functions

The Transcendental functions show the following properties:

• Continuity and Differentiability: Transcendental functions are normally continuous and also differentiable Functions.
• Growth Rates: Transcendental functions are very special kinds of functions, and most of them have very special rates of increase.
• Non-polynomial Nature: By definition, these functions cannot be expressed as finite polynomial equations.

Applications of Transcendental Functions

The applications of the Transcendental functions are as follows:

• Physics: Exponential functions model radioactive decay and population growth, while trigonometric functions describe oscillatory motions like waves.
• Engineering: Logarithmic and exponential functions are used in control systems and signal processing.
• Economics: Logarithmic functions help model economic growth and elasticity of demand.
• Biology: Exponential functions model the spread of diseases and the growth of populations.

Solved Examples Based On the Transcendental Functions

Example 1: What is the range of the function $f(x)=\left|a^x\right|$ ?
1) $[1, \infty)$
2) $R$
3) $R^{+}$
4) $[0, \infty)$

Solution:

$\begin{aligned}
& a^x>0 \forall x \equiv R \\
& \text { Thus }\left|a^x\right|=a^x \\
& \text { Hence }\left|a^x\right|>0 \quad \forall x \equiv R \\
& \text { i.e }\left|a^x\right| \in R^{+}
\end{aligned}$
Hence, the answer is the option 3.

Example 2: The number of real roots of the equation $5+\left|2^x-1\right|=2^x\left(2^x-2\right)$ is:
Solution:

$\begin{aligned}
& 5+\left|2^x-1\right|=2^x\left(2^x-2\right) \\
& 5+\left|2^x-1\right|=2^{2 x}-2 \cdot 2^x
\end{aligned}$

$\begin{aligned}
& \text { Case 1: } x \geq 0 \\
& =>5+2^x-1=2^{2 x}-2 \cdot 2^x \\
& =>2^{2 x}-2 \cdot 2^x-2^x-4=0 \\
& =>\left(2^x-4\right)\left(2^x+1\right)=0 \\
& =>2^x-4=0 \\
& \Rightarrow x=2
\end{aligned}$

Case 2: $x<0$

$\begin{aligned}
& 5+1-2^x=2^{2 x}-2 \cdot 2^x \\
& =>6=2^{2 x}-2^x \\
& \text { LHS }=+ \text { ve } \& \text { RHS }=- \text { ve (no solution) } \\
& \therefore \text { no. of solution }=1
\end{aligned}$

Hence, the answer is 1.
Example 3: Which of the following is the domain of $f(x)=\log _2(x+4)$
1) $(0, \infty)$
2) $(-2, \infty)$
3) $(-4, \infty)$

4) None of these

Solution:$\begin{aligned}
& \ln \log _2(x+4) ; x+4>0 \\
& \text { i.e } x>-4
\end{aligned}$

Hence, the answer is the option 3.
Example 4: If $f(x)=\log _c\left(\frac{1-x}{1+x}\right),|x|<1, \underset{\text { then }}{ } f\left(\frac{2 x}{1+x^2}\right)$ is equal to
1) $2 f\left(x^2\right)$
2) $2 f(x)$
3) $-2 f(x)$
4) $(f(x))^2$

Solution:

$\begin{aligned}
& f(x)=\log _e\left(\frac{1-x}{1+x}\right),|x|<1 \\
& f\left(\frac{2 x}{1+x^2}\right)=? \\
& f\left(\frac{2 x}{1+x^2}\right)=\log _e\left(\frac{1-\frac{2 x}{1+x^2}}{1+\frac{2 x}{1+x^2}}\right) \\
& =\log _e\left(\frac{\frac{1+x^2-2 x}{1+x^2}}{\frac{1+x^2+2 x}{1+x^2}}\right) \\
& =\log _e \frac{x^2-2 x+1}{x^2+2 x+1}=\log _e\left(\frac{(1-x)^2}{(x+1)^2}\right) \\
& =\log _e\left(\frac{1-x}{x+1}\right)^2
\end{aligned}$

$\begin{aligned}
& =2 \log _e\left(\frac{1-x}{x+1}\right),|x|<1 \\
& =2 f(x)
\end{aligned}$

Hence, the answer is the option 2.
Example 5: The number of solutions of the equation $\log _4(x-1)=\log _2(x-3)$ is
Solution:

$\begin{aligned}
& \log _4(x-1)=\log _2(x-3) \\
& \Rightarrow \frac{1}{2} \log _2(x-1)=\log _2(x-3) \\
& \Rightarrow \log _2(x-1)^{1 / 2}=\log _2(x-3) \\
& \Rightarrow \sqrt{x-1}=(x-3) \\
& \Rightarrow x-1=x^2+9-6 x \\
& \Rightarrow x^2-7 x+10=0 \\
& \Rightarrow(x-2)(x-5)=0 \\
& \Rightarrow x=2,5
\end{aligned}$

but $x$ can't be 2 , because in

$\log _2(x-3) \rightarrow x>3$

So, there is only one solution
Hence, the answer is 1 .