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The domain of a function is simply the set of all input values (x-values) that you can plug into the function without causing any issues. Think of it as a list of the x-values for which the function is valid and works smoothly. For example, in the function \( f(x) = 4 - \frac{2}{x} \), we need to be careful with the value of \( x \) to avoid any mathematical problems.
To find the domain, always look for values that cause issues like division by zero, taking the square root of a negative number, or logarithms of non-positive numbers.
Here, the problematic value is division by zero, which will be explained further below. Once you identify these values, exclude them from the list of acceptable x-values.

Interval notation is a shorthand way of writing down sets of numbers, especially useful for describing domains of functions.
For intervals, we use parentheses \(( , )\) to denote that endpoints are not included, and brackets \([ , ]\) to show endpoints that are included.
For example, the interval notation for all x-values less than zero and greater than zero (excluding zero) is \((-fty, 0) \cup (0, +fty)\). The union symbol \(\cup\) is used to combine sets. Using interval notation is a neat and clear way to state the domains of functions, especially when excluding specific points like zero.

Division by zero is a big no-no in mathematics. It makes any calculation invalid because dividing a number by zero does not produce a meaningful result. For the function \( f(x) = 4 - \frac{2}{x} \), the issue of dividing by zero comes from the term \( \frac{2}{x} \).
If you set the denominator to zero, which is \( x = 0 \), it makes the entire expression undefined. So, we must exclude x-values that cause this issue.
This is why in our example, we exclude 0 from the domain, making it: \((-fty, 0) \cup (0, +fty)\).
To find such problematic values, set the denominator equal to zero and solve for x. Exclude this value to maintain a valid function.