To put it simply and briefly, the scope is the values that any function can take. In order to fully explore this topic, you need to gradually disassemble the following points and concepts. First, let's understand the definition of the function and the history of its appearance.
What is a function
All the exact sciences provide us with many examples where the variables in question depend in some way 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 constant volume varies with temperature. These examples are united by the fact that all formulas have dependencies between variables, which are called functional.
A function is a concept that expresses 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 x - the argument. Thus, we can say that y is a variable dependent on the value of x. The values that x can take together arethe domain of the given function (D(y) or D(f)), and accordingly, the values of y constitute the set of function values (E(f) or E(y)). There are cases 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 notation with the formula makes sense.
There are matching or equal features. These are two functions that have equal ranges of valid values, as well as the values of the function itself are equal for all the same arguments.
Many laws of exact sciences are named similarly to situations in real life. There is such an interesting fact also about the mathematical function. There is a theorem about the limit of a function "sandwiched" between two others that have the same limit - about two policemen. They explain it this way: since two policemen are leading a prisoner to a cell between them, the criminal is forced to go there, and he simply has no choice.
Historical feature reference
The concept of a function did not immediately become final and precise, it has undergone a long way of becoming. Fermat's Introduction and Study of Plane and Solid Places, published at the end of the 17th century, first stated the following:
Whenever there are two unknowns in the final equation, there is room.
In general, this work speaks of functional dependence and its material image (place=line).
Also, around the same time, Rene Descartes was studying lines by their equations in his work "Geometry" (1637), where again the factdependence of two quantities on each other.
The very mention of the term "function" appeared only at the end of the 17th century with Leibniz, but not in its modern interpretation. In his scientific work, he considered that a function is various segments associated 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 value composed of a variable and a constant.
Euler's thoughts were also close to this:
A variable quantity function is an analytic expression made up in some way of 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 change, then the former are called functions of the latter.
Function Graph
The graph of the function consists of all points belonging to the axes of the coordinate plane, the abscissas of which take the values of the argument, and the values of the function at these points are ordinates.
The scope of a function is directly related to its graph, because if any abscissas are excluded by the range of valid values, then you need to draw empty points on the graph or draw the graph within certain limits. For example, if a graph of the form y=tgx is taken, then the value x=pi / 2 + pin, n∉R is excluded from the definition area, in the case of a tangent graph, you need to drawvertical lines parallel to the y-axis (they are called asymptotes) passing through the points ±pi/2.
Any thorough and careful study of functions constitutes a large branch of mathematics called calculus. In elementary mathematics, elementary questions about functions are also touched upon, for example, building a simple graph and establishing some basic properties of a function.
What function can be set to
Function can:
- be a formula, for example: y=cos x;
- set by any table of pairs of the form (x; y);
- immediately have a graphical view, for this the pairs from the previous item of the form (x; y) must be displayed on the coordinate axes.
Be careful when solving some high-level problems, almost any expression can be considered as a function with respect to some argument for the value of the function y (x). Finding the domain of definition in such tasks can be the key to the 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 where the function can exist. The domain of definition (x) may also be referred to as the domain of acceptable values (abbreviated as ODZ).
To correctly and quickly build a graph of functions, you need to know the domain of this function, because the appearance of the graph and fidelity depend on itconstruction. For example, to construct a function y=√x, you need to know that x can only take positive values. Therefore, it is built only in the first coordinate quadrant.
Scope of definition on 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 have a so-called complex function in front of you. It's just a combination of several 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 notation will look like this: 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 \u200b\u200bthat turn the denominator to 0. For a schematic representation, 5 well-chosen 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, the Ox and Oy axes. If the graphic image intersects with the asymptotes, then such an error will be considered the grossest.
- But what is the domain of the root? The domain of a function with a radical expression (f(x)=√(2x + 5)), containing a variable, also has its own nuances (it is only related to the root of an even degree). Asthe arithmetic root is a positive expression or equal to 0, then the root 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 a parabola, rotated by 90 degrees, located in the first coordinate quadrant.
- If we are dealing with a logarithmic function, then you should remember that there is a restriction regarding the base of the logarithm and the expression under the sign of the logarithm, in this case you can find the domain of definition as follows. We have a function: y=loga(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, you need to exclude values at which the denominator can be equal to zero. If you are familiar with the graphs of trigonometric functions, understanding their domain is a simple task.
How is working with complex functions different
Remember a few basic rules. If we are working with a complex function, then there is no need to solve something, simplify, add fractions, reduce to the lowest common denominator and extract roots. We must investigate this function because different (even identical) operations can change the scope of the function, resulting in an incorrect answer.
For example, we have a complex function: y=(x2 - 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 the function, so we do not factor the numerator and do not solve any inequalities, because the value at which the function does not exist, visible to the naked eye. In this case, x cannot take on the value 2, since the denominator cannot go to 0, the notation will look like this: D(y)=x ∉ (-∞; 2) ∪ (2; +∞).
Reciprocal functions
For starters, it's worth saying that a function can become reversible only on an interval of increase or decrease. In order to find the inverse function, you need to swap x and y in the notation and solve the equation for x. Domains of definition and domains of value are simply reversed.
The main condition for reversibility is a monotone interval of a function, if a function has intervals of increase and decrease, then it is possible to compose the inverse function of any one interval (increasing or decreasing).
For example, for the exponential function y=exthe reciprocal is the natural logarithmic function y=logea=lna. For trigonometrics, these will be functions with the prefix arc-: y=sinx and y=arcsinx and so on. Graphs will be placed symmetrically with respect to some axes or asymptotes.
Conclusions
Searching for the range of acceptable values comes down to examining the graph of functions (if there is one),recording and solving the necessary specific system of inequalities.
So, this article helped you understand what the scope of a function is for and how to find it. We hope that it will help you to understand the basic school course well.
