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When we talk about finding the domain of a function, we are trying to determine all the possible values of the independent variable (often referred to as x) that will allow the function to produce real number outputs.

Some functions have restrictions on what x can be. For example, if a function involves a square root, the value under the square root cannot be negative. In the case of rational functions, we need to avoid values that make the denominator zero, since division by zero is undefined.

Let's consider the function:






A common step involves firstly identifying these potential problematic values by solving the corresponding equation set to zero. As seen in our given exercise, setting the denominator equal to zero reveals the value that must be excluded from the domain.

Rational functions are functions that can be expressed as the ratio of two polynomials. In its general form, a rational function is written as
where P(x) and Q(x) are polynomials, and Q(x)≠0. The domain of a rational function is all real numbers except the values that make the denominator zero.

For example, consider the function:
The denominator here is ∞x-∞3. To find the domain, we set the denominator equal to zero: $$x - 3 = 0$$. Solving this, we get $$x = 3$$. This tells us that $$x = 3$$ makes the denominator zero and must be excluded from the domain. As a result, the domain of function is all real numbers except for 3.

Interval notation is a way to describe the set of all values that belong to the domain of a function. It uses parentheses and brackets:
Parentheses, ( or ), indicate that the endpoint is not included (implying an open interval).
Brackets, [ or ], indicate that the endpoint is included (implying a closed interval).

For the function
determined the domain to exclude x=3, we use parentheses to exclude this value:


The interval notation for this example is: (-∞, 3) U (3, ∞). This means that x can be any value less than 3 or any value greater than 3, but it cannot be equal to 3.

Interval notation provides a concise way to represent the domain and is useful in clearly showing which values are included and which are excluded.