Real functions | Part-2

Real Function, Real-Valued Function, and Domain of a Function

In mathematics, functions play a vital role in expressing relationships between variables. When dealing with real numbers, we often focus on real functions and real-valued functions, and understanding their domain is essential for analyzing them properly.

What is a Real Function?

A real function is a function where both the domain and the codomain are subsets of the set of real numbers (ℝ). It maps real inputs to real outputs.

Notation: f: ℝ → ℝ

Real-Valued Function

A real-valued function is a function whose range (output values) lies within the set of real numbers, although the domain might come from a different set (e.g., integers, natural numbers, etc.).

Example: A function f(n) = √n defined on natural numbers ℕ is a real-valued function.

Domain of a Function

The domain of a function refers to the complete set of input values for which the function produces valid results.

To determine the domain, we need to find all real numbers that do not lead to undefined or invalid operations such as:

• Division by zero
• Even roots of negative numbers (e.g., square root of –1)
• Logarithm of non-positive numbers

Finding the Domain of Special Functions

1. Polynomial Functions

Example: f(x) = x² + 3x + 1

Domain: All real numbers ℝ, since polynomials are defined for every real number.

2. Rational Functions

Example: f(x) = 1 / (x – 2)

Domain: All real numbers excluding x = 2, because the function becomes undefined when the denominator is zero.

Domain: ℝ – {2}

3. Square Root Functions

Example: f(x) = √(x – 4)

Condition: The expression under the square root must be non-negative.

Domain: x – 4 ≥ 0 ⇒ x ≥ 4, so the domain is [4, ∞)

4. Logarithmic Functions

Example: f(x) = log(x – 1)

Condition: The input of the logarithm must be positive.

Domain: x – 1 > 0 ⇒ x > 1, so the domain is (1, ∞)

• A real function has both domain and range within the real numbers.
• A real-valued function returns real number outputs but may have a non-real domain.
• To find the domain of a function, exclude values that result in undefined operations.