Mathematical Functions, Domain, and Range

1. What Is a Function?

In mathematics, a function is a relation that assigns exactly one output to each input. If we call the input x and the output f(x), then a function can be thought of as a machine that transforms input into output.

Simple Example:

f(x) = x + 3

In this example, the function takes the value x and adds three to it.

2. What Is the Domain?

The domain is the set of all possible inputs for which the function is defined. In other words, the domain represents the values that will not cause errors when used in the function.

Domain Example:

Consider the following function:

f(x) = 1 / x

In this function, x cannot be zero because division by zero is undefined. Therefore, the domain is all real numbers except 0.

3. What Is the Range?

The range is the set of all possible outputs that the function can produce. The range depends on the behavior of the function and its domain.

Range Example:

Consider the function:

f(x) = x^2

Since the square of any real number is always non‑negative, the range of this function is all numbers greater than or equal to 0.

4. Difference Between Domain and Range

• Domain: The allowed inputs of a function
• Range: The possible outputs of a function
• The domain is chosen by us, but the range is determined by the function’s behavior

5. Combined Example of Domain and Range

Consider the following function:

f(x) = √(x - 2)

Domain Analysis:

The expression inside the square root must be non‑negative:

x - 2 ≥ 0

So the domain is x ≥ 2.

Range Analysis:

Since the square root is never negative, the range of the function is:

f(x) ≥ 0

6. Conclusion

Understanding functions, domain, and range is essential for analyzing the behavior of any mathematical relationship. The domain defines the valid inputs, while the range shows the possible outputs. These foundational concepts are widely used in many branches of mathematics and engineering.