How the domain of a function is calculated. Exercises solved step by step.

What is the domain of a function? How to calculate the domain of a function?

Then I will explain step by step what the domain of a function is and I will teach you how to calculate it.

If you don’t know, a function is characterized because each value of x has a unique value of f(x).

However, there are times for a given value of x that no value of f(x) corresponds to it.

And what happens when no value of f(x) corresponds to it?

Well, for that value of x, the function f(x) will not exist.

Let’s see when these cases can occur.

When a function does not exist

A function will not exist when the values of x cause the following cases:

1- When a number is divided by 0:

como calcular el dominio de una función

2- When the content of a par index root is a negative number

dominio de una función

3- When the content of a logarithm is 0 or a negative number

dominio de una función

Now that you know that a function may have a value for some x values or may not exist for other x values, I’m going to explain to you what the domain of a function is.

What is the domain of a function

What is the domain of a function?

box]The domain of a function is the range of values of x for which f(x) exists, that is, the values of x, for which f(x) has a result.[/box].

It is designated as Dom f.

Seen this way it scares a little, but as we go resolving examples you are going to be much clearer.

To calculate the domain of a function, we must obtain the values of x, for which that function exists. In other words, we must find for which values of x, the function does not exist and keep the values of x where the function does exist.

The domain of a function depends a lot on the type of function.

Let’s see it:

How to calculate the domain of a function

Domain of a polynomial function

Polynomial functions are those in which neither denominators nor roots appear.

The x can appear adding, subtracting, multiplying or elevated to some exponent, as for example:

dominio de una función ejercicios resueltos

In this type of functions there is no value of x that makes f(x) not exist. Therefore, f(x) always exists.

When a function always exists, its domain is the whole set of real numbers:

dominio de una función ejercicios

How the domain of a rational function is calculated

Rational functions exist for all R, except for values that make 0 the denominator.

Therefore, to calculate the domain of a rational function, we must find the values that make 0 the denominator and take it away from R.

For example:

como se calcula el dominio de una función

This function will always exist, except when the denominator is equal to 0. Therefore, we must find that restriction that annuls the denominator.

For the function to exist, the denominator must be different from 0:

cómo se calcula el dominio de una función racional

And this restriction, is an equation of first degree, from where we must clear the x:

dominio de una función

When x=1, the denominator will be 0. Therefore, for f(x) to exist, x has to be different from 1 and that is the value to be removed from R:

dominio de funciones

The domain is all R except the set formed by the number 1.

Let’s see another example:

como calcular dominio

As before, this function will exist as long as the denominator is not 0. Therefore, we calculate the values that make 0 the denominator:

cálculo del dominio de una función

That is to say, the function will exist whenever x is different from 2 and 3, therefore the domain is all R except 2 and 3:

ejercicios de dominio de funciones

How to calculate the domain of an irrational function

Irrational functions are those in which a root appears.

Those of odd index always exist.

Those with even indexes exist as long as their content is equal to or greater than zero.

For example:

como calcular dominio de una función

It is an even index root, therefore, it exists as long as its content is greater than or equal to 0:

el dominio de una función

From this inequality, we clear x and we are left:

cuál es el dominio de una función racional

Therefore, the function will exist as long as x is greater than or equal to -2. Its domain is::

como determinar el dominio de una función

Another example:

cual es el dominio de una función de primer grado

It is an odd index root, then it always exists and its domain is all R:

como hacer el dominio de una función

How to calculate the domain of a logarithmic function

Logarithmic functions with those where x is within a logarithm.

They exist as long as the content of the logarithm is not 0 or a negative number, that is, they will exist as long as the content of the logarithm is greater than 0.

For example:

dominio ejercicios resueltos

This function will exist as long as the content of the logarithm is greater than 0:

como se halla el dominio de una función

From this inequality, we clear the x and we are left:

como hallar el dominio de una función ejercicios resueltos

The function will exist as long as x is greater than 3. Therefore its domain is:

cual es el dominio de una función polinomial

How to graphically identify the domain of a function?

Graphically the domain of a function are the values of x for which the graph of the function appears drawn above. If above a value of x there is nothing, that value of x does not belong to the domain.

The domain is then always looked at on the x-axis.

For example, we have the graph of the following function, what is its domain?:

dominio de una función logarítmica ejercicios resueltos

The domain is the range of values of x, for which the function is drawn above. As we can see, from 1 to 4, the function is drawn. In no other value of x more:

calcular dominios de funciones

Therefore, the domain of this function is:

ejemplo de dominio en matemáticas

Therefore, as we have already seen, the domain of a function is the range of values for which the function exists.