What is the domain of a function and how do you find it?

What is the Domain of a Function?

In mathematics, the domain of a function is the set of all possible inputs (or independent variables) that can be used to produce a valid output (or dependent variable). In other words, it's the range of values that the input variable can take. For example, if you have a function like $f(x) = \sqrt{x}$ , the domain would be all non-negative real numbers, because you can't take the square root of a negative number in the real number system.

How to Find the Domain of a Function?

To find the domain of a function, you need to consider the following:

Allowable Operations: The domain must include values that allow for the mathematical operations defined in the function. For instance, you can't divide by zero, so if a function has a denominator, zero cannot be in the domain.
Real Numbers vs. Complex Numbers: If the function involves square roots, square numbers, or other operations that aren't defined for negative numbers in the real number system, then the domain must be restricted to non-negative numbers.
Explicit Restrictions: Sometimes, the function might have explicit restrictions on the domain, such as "x > 0" or "x ≠ 3".

Here's how you might find the domain of a function like $f(x) = \sqrt{x + 3} / (x - 2)$ :

The square root function requires its input to be non-negative, so $x + 3 \geq 0 \Rightarrow x \geq -3$ .
The denominator cannot be zero, so $x - 2 \neq 0 \Rightarrow x \neq 2$ .
Combining these two conditions, the domain of $f(x)$ is all real numbers greater than or equal to -3, but not equal to 2, which can be written as $(-3, 2) \cup (2, \infty)$ .