In a soccer stadium there is a soccer field that has a certain size. There are various fixed lengths, for example the length of the goal line or the length of the penalty area.
The sideline is about 105 meters long. It is now important for football teams that the sideline must be between 100 and 110 meters. Any values below or above are not allowed.
You have already set up your first definition area. Determining the domain of definition is often the task of a curve discussion and is explained in more detail in this explanation.
Determining the domain of definition – repetition of sets of numbers
At the beginning it would be an advantage to briefly repeat everything about the different sets of numbers.
A lot is that Linking of different objects, like an apple tree and a walnut tree can generally be counted among the trees. This is also the case with numbers.
There are different numbers that you get sets of numbers can summarize.
The smallest form are the so-called natural numbers. Next there is the whole numbersrightnational numbers, irrational numbers and the real numbers in school mathematics.
The table summarizes all important numbers.
Set of numbersIndicationNatural numbers\( \mathbb{N} = \{ 0, 1, 2, 3, …\}\)Whole numbers\( \mathbb{Z} = \{…, -3, -2, – 1, 0, 1, 2, 3, … \} \)Rational numbers\( \mathbb{Q} = \{ \frac{m}{n} | n \in \mathbb{Z}, n \neq 0\}\)Real numbers\(\mathbb{R} = \mathbb{Q} \cup \mathbb{I} \)
You can also show the whole thing in one picture.
Figure 1: Determining the domain of definition – sets of numbers
the natural numbers \(\mathbb{N}\) are all positive integers including zero.
For the whole numbers \(\mathbb{Z}\) come the integers negative Pay with it.
Both rightnational numbers \(\mathbb{Q}\) then all numbers that can be represented as a fraction are included. If this is no longer possible, these numbers are again included in the set of irrational numbers \(\mathbb{I}\) is given.
Finally, there are a few special numbers, such as the circle number \( \pi \) , Euler’s number \( e\) or \(3 \cdot \sqrt{2}\) , which belong to the real numbers \(\mathbb{R}\) that have an infinite number of decimal places.
Curve Discussion – Determine domain of definition
A function \(f\) is generally an assignment, whereby each element \(x\) from a domain \(\mathbb{D}\) is assigned an element from \(y\). For example, a function can consist of the points \(P = (0,0)\), \(Q = (1,1)\) and \(R = (2,2)\).
Determine the domain of a function
However, the so-called definition set or the definition range can be restricted.
Of the domain of definition \( \pmb{\mathbb{D}}\) or the definition set describes which x-values can be used in a function \(f(x)\).
If a domain is asked for via a specific function such as \(f(x)\), this can also be written as \( \pmb{\mathbb{D_f}}\).
The domain of definition thus answers the question of which x-values may be used in the function.
For example, you are given the following function:
\
In addition, the definition set should hold:
\
So that means you only for this given x-values can create the function. The values for the y-coordinates are therefore:
- \(f(1) = 3 \times 1 = 3\)
- \(f(2) = 3 \times 2 = 6\)
- \(f(3) = 3 \times 3 = 9\)
- \(f(4) = 3 \times 4 = 12\)
The y-values are part of the set of valueswhich you will soon meet.
It is therefore possible that in a task the domain of definition, and thus the possible values to be used, is restricted at will.
Determine range of values
The restriction of a domain of definition often has a direct impact on the so-called domain of values.
Of the range of values \( \pmb{\mathbb{W}}\) or also the set of values of a function \(f(x)\) are the y-values that a function can assume (depending on the respective domain).
If the domain of definition is restricted, the value range of this function is also reduced.
With most functions, the domain of definition is not limited and everything from \(- \infty \, to + \infty \) can be used.
That also means for them set of valuesthat all values may come out of your set of numbers.
However, if a restriction is made in the definition set (as in the previous chapter), the value set can also be restricted.
This is also the function
\
The definition set is limited here. It should be possible to use all points on the function for the x-value 1 to 4. The set of values is now also limited.
The same applies to the set of values:
\ \]
This means that this time all x-values from 1 to 4 can be used and all y-values between 1 and 4 that are combined as points on this line. Here also the integers in the interval are given as dots.
Figure 2: Domain of definition for a straight line
interval notation
The interval notation is also important for the domain of definition. This describes which numbers can be chosen within specified limits. The limits of these intervals may or may not belong there. So can the following interval
\ \]
also be written in this form.
\
However, if both limits are no longer included, for example, the 3 and the 5 can no longer be used.
\ 3; 5 or \
This means that you use the interval notation if not all numbers from a set of numbers can be used with a few exceptions, but explicitly any range of numbers.
However, you can find more information on this in the explanation of intervals.
Difference between solution set and definition set
The relationship between the solution set and the definition set is that at the beginning of a task it is defined which numbers can be used. If an equation is then solved for a variable, you get a so-called solution set. This indicates for which x-values an equation is fulfilled.
It is then still necessary to check whether these values are determined in the definition set. If this is the case, you can use it to determine the solution set. If not, these values must be taken from the solution set.
Assuming you gave two pieces of information:
- Equation \(x = -2\)
- definition set are the natural numbers \(\mathbb{N}\)
1st step (solve equation):
This means that if the equation is to be solved, then you need the number -2 for the x-value, since then the following applies:
\
2nd step (comparison of solution and definition set):
However, this number is not defined in the domain of definition, so the so-called empty set applies to the solution set.
\
Other case:
However, if the definition set whole numbersor the rational or real numbers, then the solution set is:
\
Determine the maximum domain
There are different domains of definition for different functions.
You always start from the largest set of numbers from what you have known so far. The rational numbers \( \mathbb {R}\) are generally used here, but feel free to choose the one that suits you.
If you are now asked to determine the domain, then that is maximum domain meant.
Whole Rational Functions and Exponential Function
In principle, it is possible to use any value from the real numbers for an integral function.
For the whole rational functions like the linear function, but also the quadratic function and functions greater than degree two applies to the definition set \( \mathbb{D} = \mathbb{R}\). Also for them exponential function the same domain of definition applies.
In this case, the basic set is not restricted, nor is the set of values for the linear functions restricted. Only for the set of values \(\mathbb{W}\) for quadratic functions is it to be noted that these…
- from the y-value of the vertex to \(+ \infty\) for \(a > 0\)
- from the y-value of the vertex to \(- \infty \) for \(a < 0\)
enough.
You can find more information about the quadratic functions and how to calculate the apex, or what applies to the set of values, in the following explanations:
Incidentally, the fact that you can use whole real numbers is also the case with the exponential function \( e^x \). Again, there is no restriction of the domain of definition.
Determine the domain of definition for each of the following examples. (Hint: you only need one solution).
\
\
\
These are real numbers, since there are no restrictions and they run from \(- \infty\) to \( + \infty \).
\
Broken rational functions – definition gaps
With fractional rational functions, it is crucial that the denominator can never be 0, since, as is well known, you cannot divide by 0. After all, it is not possible to distribute a cake to 0 people.
For the Domain of a Fractional Rational Function of the form \( \frac{a}{x} \) it is crucial that the denominator should not be equal to 0. To do this, the denominator is compared with 0 to determine the domain.
This means that you look at the denominator of the function and set it equal to 0. The values that are determined – sometimes using a linear equation, or using the midnight formula or substitution – can be found in the domain of definition.
So, for the simplest case, a broken rational function is considered:
\
In this case x cannot take the value 0. This means that the real numbers are always used, but without this value.
\
You can try these steps yourself for a less intuitive function.
Find the domain of the following function.
\
Step 1:
Basically, the denominator of the function must be considered and set to 0. You can ignore the counter for the domain.
\begin{align} 0 &= k(x) \\ 0 &= \frac{5}{x^2 + 3x – 2x – 6} \end{align}
Step 2:
You can first summarize the term and then use it for the midnight formula.
\begin{align} 0 &= x^2 + 3x – 2x – 6 \\ &= x^2 + x – 6\end{align}
The midnight formula is:
\
\begin{align} x_{1,2} &= \frac{-1 \pm \sqrt{1^2 – 4 \cdot 1 \cdot (-6)}} {2 \cdot 1} \\ &= \ frac{ -1 \pm \sqrt{25}}{2} \\ &= \frac{-1 \pm 5}{2} \\ x_1 &= \frac{-1 + 5}{2} \\ & = 2 \\ x_2 &= \frac{-1 – 5}{2} \\ &= -3 \end{align}
This means that these two values should not be included in the domain, so you write them as follows:
\
(Natural) logarithmic function
As a start you are given a logarithmic function, where you may already notice a small peculiarity in terms of the domain of definition.
Figure 3: Domain of log function and shifted ln function
In this graph, all x-values are greater than 0 for the log function. Negative x-values are also possible for the natural logarithm. This is explained below.
Of the Domain of a logarithmic function of the form \(y = \log{b}\) x as an inverse function of the exponential function with the form \(y = b^x\), all…