The domain of a function can be written either explicitly or implicitly through the equation used to define the function. The implicit domain is the set of all real numbers for which the equation is defined, while an explicitly defined domain is the one given together with the function. For example, the function given by:

As can be seen, the function has an explicit domain, since the function is only defined for thex that are between 4 and 5.

On the other hand, if we express the function as follows:

The domain for this case is expressed implicitly, i.e. the domain is defined for the values of x for which the function exists, and for this we must perform a general analysis to see which are the values of x for which the function exists.

For the case of the rational function, it exists for any value of x, except for x = 2, since when substituting the 2 and -2 in x² gives us 4, and then subtracting with 4 gives us zero, which implies that division by zero is not defined.

The way to express this domain implicitly is:

The domain is the set:

{𝑥: 𝑥 ≠ ±2}

Which means that the domain is all real numbers except 2 and -2.

Calculation of the domain and path of a function

To calculate the domain and path of a real function, I am going to propose to find the domain and path of the following function:

The first thing we must take into account is that what is inside a square root must be greater than or equal to zero, which implies that:

We clear the variable x:

That x is greater than or equal to one, means that x is the set of values that have to be greater than or equal to 1, i.e. the domain is the set that spans the following interval [1;∞)

The graph of the domain is:

To find the path or range of the function we observe that the images y = f(x) are never negative, which means that the range or path is the set [0;∞), i.e. the graph on the Y-axis goes from zero to positive infinity.

Bibliographic Reference

Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006