Transcendental functions are functions that cannot be expressed as a finite combination of algebraic operations (for example, addition, subtraction, multiplication, division, and roots).

Types of Transcendental Functions

There are 3 main types of transcendental functions used in data science, namely:

1. Exponential Functions:

• Form:

  $$f(x) = a \cdot e^{bx},$$
  • Where $a$ and $b$ are constants.

• Use case: growth and decay models, compound interest.

2. Logarithmic Functions:

• Form: $$f(x) = \log_b (x),$$ $$f(x) = ln(x);$$
• Use case: measuring pH levels, earthquake magnitudes (Richter scale).

3. Trigonometric Functions:

• Form:

  $$f(x) = \sin(x),$$ $$f(x) = \cos(x),$$ $$f(x) = \tan(x);$$

• Use case: signal processing, physics, engineering.

After you've run the code, the following observations should be evident:

• Exponential Function $f(x) = e^x$: exhibits rapid growth for positive $x$ and approaching zero for negative $x$. The graph includes an arrow indicating continuous growth towards infinity;

• Logarithmic Function $f(x) = \log_2(x)$: defined only for $x > 0$, growing slowly as $x$ increases. The arrow at the right end represents the function's unbounded nature;

• Sine and Cosine Functions $f(x) = sin(x), cos(x)$: constitutes periodic oscillations between -1 and 1. The arrows at both ends highlight their infinite continuity;

• Tangent Function $f(x) = tan(x)$: exhibits vertical asymptotes at $x = -\frac{\pi}{2}, \frac{\pi}{2}, ...$, where the function approaches infinity. The graph ensures no abrupt stops and smoothly approaches asymptotes.