In this section of the work, we will discuss more detailed functions where both the domain and codomain are real numbers (R). These functions will be referred to as Real-Valued Functions.

2.3.1. Polynomial Functions
Polynomial functions are expressed in the form

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … as follows.
Here:

• aᵢ ∈ R: This is the coefficient of x, and each coefficient is a real number.
• n ∈ N: The highest power of x, n, also represents the degree of the polynomial.
• Degree 0: When n = 0, the polynomial becomes a constant function.
• Degree 1: When n = 1, the polynomial becomes a linear equation.

2.3.2. Exponential Functions
Exponential functions are expressed in the form: 𝑎ˣ.

Here:

• a > 0: a is a constant positive real number.
• x ∈ R: The exponent is any real number.

In quantum computing:

• a = 2 is taken.
  For example, 𝑓(𝑥) = 2ˣ. In this case, the function represents exponential growth.
• Another common value is a = e,
  which is approximately 2.718, the base of the natural logarithm. In this case, the function is expressed as 𝑓(𝑥) = eˣ and is called the natural exponential function.

2.3.3. Logarithmic Function
Logarithmic functions are the inverses of exponential functions. The logarithmic function with base 2 is expressed as:

• 𝑓(𝑥) = log₂𝑥, where 𝑦 is the value of the function and, equivalently, we get the relation 𝑥 = 2^𝑦 

2.3.3.1. Properties of the Logarithmic Function:

• The domain of the function 𝑓(𝑥) = log₂𝑥 is restricted to positive real numbers (ℝ⁺ = (0, +∞)). This is because the logarithm of 0 and negative numbers is not defined in the set of real numbers.
• Logarithmic values can be calculated using the following formula:
  log₂𝑥 = log₁₀𝑥 / log₁₀2.