Domain of a Function


The domain of a function is a set of all points in the x-axis that one can be set in a function and get a defined result.

Example 1

In the function:


One can set any number from minus infinity to infinity. Therefore, the domain of the function is all x. We write it like this:

-\infty< x< \infty

Example 2

In the function:


הביטוי בתוך השורש לא יכול להיות שלילי, ולכן נדרוש שיתקיים:

The expression inside the square root cannot be negative, so we require

x+2\geq 0

x\geq -2

We got that one can set only the number 2 or larger in the function. Therefore, the domain of ​​the function is

x\geq -2

This is how the domain of functions looks:

In one variable functions, the domain is a segment (finite or infinite) in the x-axis, or all the x-axis (and then it is stated that the domain is all x).

In two variables functions, the domain is in the XY plane, finite or infinite. It can be a finite number of points, a line or several lines on a plane, a closed shape like a circle, triangle, square and more or an endless open form.

In three variables functions the domain is in the space XYZ. It can be a closed domain like a sphere or open like an endless cylinder.

If the domain equation is an equation, then the domain includes only the points on the equation. However, if the domain is defined with inequality, then it includes collection of points on one side of the equation. If the sign of the inequality includes the equal sign, then the domain includes both the points on the equation and the points on one side.

How to find domain of a given function

The truth is, it’s pretty simple. All you have to do is go through these rules:

  • If there is a fraction in the function, then its denominator should be different from zero.
  • If the function has an even root (square, forth and so on), then the expression inside the root should be non-negative, i.e. greater or equal to zero. Note that with an odd root (like a third root) there is no problem and it does not limit the x values.
  • If there is a log function, then the expression inside it must be greater than zero, i.e. neither zero nor negative number. Remember that ln is a log (with a base e). Also note that by setting a log, the log base must also be greater than zero, and different than one, but there will usually be no variable in the log base, but some number, which meets these conditions.
  • If there is an arcsin or arccos function, then the expression inside the arcsin or arccos should be between 1 and (-1), including these points.

If no one appears in the function – no denominator, no root, and no log – then the domain is all x, and if several things appear, then the final domain is the intersection (“and” relation) between all results.

Press here for exercises and solutions in one variable function domain

Press here for exercises and solutions in multivariable function domain

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