1. What a Real-Valued Function Means
A real-valued function is any function whose outputs are real numbers. The inputs may come from various sets, but every output must be a real number.
In short: the function may take different types of inputs, but the results always lie in \( \mathbb{R} \).
2. Formal Definition
If a function maps elements from a set A to the set of real numbers, it is called a real-valued function.
2.1. Definition
f : A \to \mathbb{R}
f(x) \in \mathbb{R} \; \text{for every } x \in A
3. Examples of Real-Valued Functions
Here are some simple and common real-valued functions:
3.1. Example 1 — Polynomial Function
\( f(x) = x^2 - 3x + 2 \)
All outputs are real numbers.
3.2. Example 2 — Absolute Value Function
\( f(x) = |x| \)
The absolute value is always a real number.
3.3. Example 3 — Square Root Function (Restricted Domain)
\( f(x) = \sqrt{x} \)
Outputs are real only when x ≥ 0.
3.4. Example 4 — Trigonometric Functions
\( f(x) = \sin x \)
Outputs stay between -1 and 1, which are real numbers.
4. Real-Valued Functions from Various Input Sets
Inputs don't have to be real numbers. The only condition is that outputs must be real.
4.1. Examples
- A function from natural numbers to real numbers.
- A function from integers to real numbers.
- A function defined on vectors but returning a real value (like length).
5. Range of a Real-Valued Function
The range of a real-valued function is always a subset of \( \mathbb{R} \). The exact range depends on the rule of the function.
5.1. Illustration
For \( f(x) = x^2 \), the range is:
[0, \infty)
6. Why Real-Valued Functions Matter
Real-valued functions appear in almost every part of mathematics. They describe physical quantities, measurements, distances, rates, and many real-world phenomena. Their outputs being real numbers makes them easy to plot, analyze, and apply in calculations.