To put it simply and briefly, the domain of definition is those values that a function can take. In order to fully explore this topic, it is necessary to gradually analyze the following points and concepts. To begin, let's look at the definition of a function and the history of its appearance.
What is a function?
All the exact sciences provide us with many examples when the variables in question somehow depend on one another. For example, the density of a substance is completely determined by its mass and volume. The pressure of an ideal gas at a constant volume varies with temperature. These examples are united by the fact that all formulas have dependencies between variables, which are called functional.
Function is a concept expressing the dependence of one quantity on another. It has the form y = f (x), where y is the value of the function, which depends on the x - argument. Thus, we can say that y is a variable depending on the value of x. The values that x can take together constitute the domain of definition of the given function (D (y) or D (f)), and accordingly, the values of y comprise the set of values of the function (E (f) or E (y)). There are times when a function is given by some formula. In this case, the domain of definition consists of the value of such variables, in which the record with the formula makes sense.
There are matching or equal functions. These are two functions for which the ranges of admissible values are equal, and also the values of the function itself are equal for all the same arguments.
Many laws of the exact sciences are called similarly to situations in real life. There is such an interesting fact about the mathematical function as well. There is a theorem on the limit of a function "sandwiched" between two others having the same limit - on two policemen. They explain it this way: since two policemen are leading a prisoner into a cell between themselves, the criminal is forced to go there, and he simply has no choice.
Feature History
The concept of function did not immediately become final and exact; it underwent a long way of becoming. First, in the work of Fermat “Introduction and study of flat and bodily places”, which was published at the end of the 17th century, the following was said:
Whenever there are two unknown quantities in the final equation, there is a place.
In general, this work speaks of functional dependence and its material image (place = line).
Also at about the same time, Rene Descartes studied the lines according to their equations in his work "Geometry" (1637), where again the fact of the dependence of the two quantities on each other was traced.
The very mention of the term "function" appeared only at the end of the 17th century by Leibniz, but not in its modern interpretation. In his scientific work, he considered that a function is various segments connected with a curved line.
But already in the 18th century, the function began to be defined more correctly. Bernoulli wrote the following:
A function is a quantity made up of a variable and a constant.
Euler's thoughts were also close to this:
The function of a variable quantity is an analytic expression composed in some way from this variable quantity and numbers or constant quantities.
***
When some quantities depend on others in such a way that when the latter change, they themselves undergo a change, the former are called the functions of the latter.
Function graph
The function graph is made up of all points belonging to the axes of the coordinate plane, the abscissas of which take values of the argument, and the values of the function at these points are ordinates.
The domain of the function is directly related to its graph, because if any abscissas are excluded by the region of acceptable values, then you need to draw empty points on the graph or draw a graph within certain limits. For example, if you take a graph of the form y = tgx, then the value x = pi / 2 + pi * n, n∉R is excluded from the definition domain, in the case of the tangent graph, you need to draw vertical lines parallel to the Oy axis (they are called asymptotes) passing through points ± pi / 2.
Any thorough and thorough research of functions constitutes a large section of mathematics called mathematical analysis. In simple mathematics, they also touch upon elementary questions concerning functions, for example, constructing a simple graph and establishing some basic properties of a function.
What function can be set
The function may:
- be a formula, for example: y = cos x;
- set by any table of pairs of the form (x; y);
- immediately have a graphic view, for this pair from the last item of the form (x; y) should be displayed on the coordinate axes.
Be careful when solving some high-level tasks, almost any expression can be considered as a function relative to some argument for the value of the function y (x). Finding a scope in such tasks can be the key to a solution.
What is the scope for?
The first thing you need to know about a function in order to study or build it is its scope. The graph should contain only those points at which the function can exist. The domain of definition (x) may also be referred to as the region of acceptable values (abbreviated as DLD).
To correctly and quickly build a graph of functions, you need to know the domain of definition of this function, because the appearance of the graph and the accuracy of the construction depend on it. For example, to construct the function y = √x, you need to know that x can take only positive values. Therefore, it is built only in the first coordinate quarter.
The domain of definition by the example of elementary functions
In its arsenal, mathematics has a small number of simple, defined functions. They have a limited scope. The solution to this issue will not cause difficulties even if you are faced with the so-called complex function. This is just a combination of a few simple ones.
- So, the function can be fractional, for example: f (x) = 1 / x. Thus, the variable (our argument) is in the denominator, and everyone knows that the denominator of a fraction cannot be equal to 0, therefore, the argument can take any value except 0. The record will have the following form: D (y) = x∈ ( -∞; 0) ∪ (0; + ∞). If there is some expression with a variable in the denominator, then you need to solve the equation for x and exclude the values that turn the denominator to 0. For the schematic image, 5 correctly selected points are enough. The graph of this function will be a hyperbola with a vertical asymptote passing through the point (0; 0) and in combination with the axis Ox and Oy. If the graphic image intersects with the asymptotes, then such an error will be considered gross.
- But what is the scope of the root? The domain of definition of a function with a radical expression (f (x) = √ (2x + 5)) containing a variable also has its own nuances (it relates only to the root of even degree). Since the arithmetic root is a positive or equal to 0 expression, the radical expression must be greater than or equal to 0, we solve the following inequality: 2x + 5 ≥ 0, x ≥ -2.5, therefore, the domain of this function: D (y) = x ∈ (-2.5; + ∞). The graph is one of the branches of the parabola, rotated 90 degrees, located in the first coordinate quarter.
- If we are dealing with a logarithmic function, then we should remember that there is a limitation with regards to the base of the logarithm and the expression under the sign of the logarithm, to find the domain in this case as follows. We have a function: y = log a (x + 7), we solve the inequality: x + 7> 0, x> -7. Then the domain of this function is D (y) = x ∈ (-7; + ∞).
- Also pay attention to trigonometric functions of the form y = tgx and y = ctgx, since y = tgx = sinx / cos / x and y = ctgx = cosx / sinx, therefore, it is necessary to exclude values at which the denominator can be equal to zero. If you are familiar with the graphs of trigonometric functions, figuring out their scope is a simple task.
What is the difference between working with complex functions
Remember a few basic rules. If we work with a complex function, then you do not need to solve, simplify, add fractions, lead to the lowest common denominator and extract the roots. We must investigate this function, because different (even identical) operations can change the scope of the function, which will lead to an incorrect answer.
For example, we have a complex function: y = (x 2 - 4) / (x - 2). We cannot reduce the numerator and denominator of the fraction, since this is possible only if x ≠ 2, and this is the task of finding the domain of definition of the function, therefore, we do not factor the numerator and do not solve any inequalities, because the value at which the function does not exist seen with the naked eye. In this case, x cannot take the value 2, since the denominator cannot go to 0, the entry will look like this: D (y) = x ∉ (-∞; 2) ∪ (2; + ∞).
Mutually Inverse Functions
To begin with, it is worth saying that a function can become reversible only in the period of increase or decrease. In order to find the inverse function, you need to swap x and y in the record and solve the equation for x. The domains of definition and the domain of meaning are simply swapped.
The main condition for reversibility is the monotonic interval of the function, if the function has intervals of increase and decrease, then we can compose the inverse function of any one interval (increasing or decreasing).
For example, for the exponential function y = e x the inverse will be the natural logarithmic y = log e a = lna. For trigonometric ones, these will be functions with the prefix arc-: y = sinx and y = arcsinx and so on. The graphs will be arranged symmetrically with respect to some axes or asymptotes.
conclusions
The search for the region of admissible values is reduced to studying the graph of functions (if any), writing and solving the necessary specific system of inequalities.
So, this article helped you understand why a function definition area is needed and how to find it. We hope that it will help you to understand the basic school course well.