How to find the domain of a function? Middle school students often have to deal with this task.
Parents should help their children understand this issue.
Specifying a function.
Let us recall the fundamental terms of algebra. In mathematics, a function is the dependence of one variable on another. We can say that this is a strict mathematical law that connects two numbers in a certain way.
In mathematics, when analyzing formulas, numeric variables are replaced by alphabetic symbols. The most commonly used are x (“x”) and y (“y”). The variable x is called the argument, and the variable y is called the dependent variable or function of x.
Exist various ways setting variable dependencies.
Let's list them:
- Analytical type.
- Tabular view.
- Graphic display.
The analytical method is represented by the formula. Let's look at examples: y=2x+3, y=log(x), y=sin(x). The formula y=2x+3 is typical for linear function. Substituting into the given formula numeric value argument, we get the value of y.
The tabular method is a table consisting of two columns. The first column is allocated for the X values, and in the next column the data of the player is recorded.
The graphical method is considered the most visual. A graph is a display of the set of all points on a plane.
To construct a graph use Cartesian system coordinates The system consists of two perpendicular lines. Identical unit segments are laid on the axes. The countdown is made from center point intersection of straight lines.
The independent variable indicates horizontal line. It is called the abscissa axis. The vertical line (y-axis) displays the numerical value of the dependent variable. Points are marked at the intersection of perpendiculars to these axes. Connecting the points together, we get solid line. It is the basis of the schedule.
Types of variable dependencies
Definition.
In general, the dependence is presented as an equation: y=f(x). From the formula it follows that for each value of the number x there is certain number u. The value of the game, which corresponds to the number x, is called the value of the function.
All possible values that the independent variable acquires form the domain of definition of the function. Accordingly, the entire set of numbers of the dependent variable determines the range of values of the function. The domain of definition is all values of the argument for which f(x) makes sense.
Initial task for research mathematical laws consists in finding the domain of definition. This term must be correctly defined. IN otherwise all further calculations will be useless. After all, the volume of values is formed on the basis of the elements of the first set.
The scope of a function is directly dependent on the constraints. Limitations are caused by the inability to perform certain operations. There are also limits to the use of numerical values.
In the absence of restrictions, the domain of definition is the entire number space. The infinity sign has a horizontal figure eight symbol. The entire set of numbers is written like this: (-∞; ∞).
IN certain cases the data array consists of several subsets. The scope of numerical intervals or spaces depends on the type of law of parameter change.
Here is a list of factors that influence the restrictions:
- inverse proportionality;
- arithmetic root;
- exponentiation;
- logarithmic dependence;
- trigonometric forms.
If there are several such elements, then the search for restrictions is divided for each of them. The biggest problem represents identification critical points and intervals. The solution to the problem will be to unite all numerical subsets.
Set and subset of numbers
About sets.
The domain of definition is expressed as D(f), and the union sign is represented by the symbol ∪. All numerical intervals enclosed in brackets. If the boundary of the site is not included in the set, then a semicircular bracket is placed. Otherwise, when a number is included in a subset, square brackets are used.
Inverse proportionality is expressed by the formula y=k/x. The function graph is a curved line consisting of two branches. It is commonly called a hyperbole.
Since the function is expressed as a fraction, finding the domain of definition comes down to analyzing the denominator. It is well known that in mathematics division by zero is prohibited. Solving the problem comes down to equalizing the denominator to zero and finding the roots.
Here's an example:
Given: y=1/(x+4). Find the domain of definition.
- We equate the denominator to zero.
x+4=0 - Finding the root of the equation.
x=-4 - Define the set of all possible values argument.
D(f)=(-∞ ; -4)∪(-4; +∞)
Answer: The domain of the function is all real numbers except -4.
The value of a number under the square root sign cannot be negative. In this case, defining a function with a root is reduced to solving an inequality. The radical expression must be greater than zero.
The area of determination of the root is related to the parity of the root indicator. If the indicator is divisible by 2, then the expression makes sense only if it positive value. Odd number indicator indicates the admissibility of any meaning of the radical expression: both positive and negative.
Inequalities are solved in the same way as equations. There is only one difference. After multiplying both sides of the inequality by a negative number the sign should be reversed.
If the square root is in the denominator, then an additional condition must be imposed. The number value must not be zero. Inequality moves into the category of strict inequalities.
Logarithmic and trigonometric functions
The logarithmic form makes sense when positive numbers. Thus, the domain of definition logarithmic function similar to the square root function, except zero.
Let's consider an example of a logarithmic dependence: y=log(2x-6). Find the domain of definition.
- 2x-6>0
- 2x>6
- x>6/2
Answer: (3; +∞).
The domain of definition of y=sin x and y=cos x is the set of all real numbers. There are restrictions for tangent and cotangent. They are associated with division by the cosine or sine of an angle.
The tangent of an angle is determined by the ratio of sine to cosine. Let us indicate the angle values at which the tangent value does not exist. The function y=tg x makes sense for all values of the argument except x=π/2+πn, n∈Z.
The domain of definition of the function y=ctg x is the entire set of real numbers, excluding x=πn, n∈Z. If the argument is equal to the number π or a multiple of π, the sine of the angle equal to zero. At these points (asymptotes) the cotangent cannot exist.
The first tasks to identify the domain of definition begin in lessons in the 7th grade. When first introduced to this section of algebra, the student should clearly understand the topic.
It should be noted that this term will accompany the student, and then the student, throughout the entire period of study.
First, let's learn how to find domain of definition of the sum of functions. It is clear that such a function makes sense for all such values of the variable for which all the functions that make up the sum make sense. Therefore, there is no doubt about the validity of the following statement:
If the function f is the sum of n functions f 1, f 2, …, f n, that is, the function f is given by the formula y=f 1 (x)+f 2 (x)+…+f n (x), then the domain of definition of the function f is the intersection of the domains of definition of the functions f 1, f 2, ..., f n. Let's write this as .
Let's agree to continue to use entries similar to the last one, by which we mean written inside a curly brace, or the simultaneous fulfillment of any conditions. This is convenient and quite naturally resonates with the meaning of the systems.
Example.
The function y=x 7 +x+5+tgx is given, and we need to find its domain of definition.
Solution.
The function f is represented by the sum of four functions: f 1 - power function with exponent 7, f 2 - power function with exponent 1, f 3 - constant function and f 4 - tangent function.
Looking at the table of areas for defining the main elementary functions, we find that D(f 1)=(−∞, +∞) , D(f 2)=(−∞, +∞) , D(f 3)=(−∞, +∞) , and the domain of definition of the tangent is the set of all real numbers except numbers 
.
The domain of definition of the function f is the intersection of the domains of definition of the functions f 1, f 2, f 3 and f 4. It is quite obvious that this is the set of all real numbers, with the exception of the numbers .
Answer:
the set of all real numbers except .
Let's move on to finding domain of definition of a product of functions. For this case, a similar rule applies:
If the function f is the product of n functions f 1, f 2, ..., f n, that is, the function f is given by the formula y=f 1 (x) f 2 (x)… f n (x), then the domain of definition of the function f is the intersection of the domains of definition of the functions f 1, f 2, ..., f n. So, .
This is understandable, in the indicated area all product functions are defined, and hence the function f itself.
Example.
Y=3·arctgx·lnx .
Solution.
The structure of the right-hand side of the formula defining the function can be considered as f 1 (x) f 2 (x) f 3 (x), where f 1 is a constant function, f 2 is the arctangent function, and f 3 is a logarithmic function with base e.
We know that D(f 1)=(−∞, +∞) , D(f 2)=(−∞, +∞) and D(f 3)=(0, +∞) . Then 
.
Answer:
The domain of definition of the function y=3·arctgx·lnx is the set of all real positive numbers.
Let us separately focus on finding the domain of definition of a function given by the formula y=C·f(x), where C is some real number. It is easy to show that the domain of definition of this function and the domain of definition of the function f coincide. Indeed, the function y=C·f(x) is the product of a constant function and a function f. The domain of a constant function is the set of all real numbers, and the domain of a function f is D(f) . Then the domain of definition of the function y=C f(x) is 
, which is what needed to be shown.
So, the domains of definition of the functions y=f(x) and y=C·f(x), where C is some real number, coincide. For example, the domain of the root is , it becomes clear that D(f) is the set of all x from the domain of the function f 2 for which f 2 (x) is included in the domain of the function f 1 .
Thus, domain of definition of a complex function y=f 1 (f 2 (x)) is the intersection of two sets: the set of all such x that x∈D(f 2) and the set of all such x for which f 2 (x)∈D(f 1) . That is, in the notation we have adopted 
(this is essentially a system of inequalities).
Let's look at some example solutions. We will not describe the process in detail, as this is beyond the scope of this article.
Example.
Find the domain of definition of the function y=lnx 2 .
Solution.
The original function can be represented as y=f 1 (f 2 (x)), where f 1 is a logarithm with base e, and f 2 is a power function with exponent 2.
Turning to known areas definitions of the basic elementary functions, we have D(f 1)=(0, +∞) and D(f 2)=(−∞, +∞) .
Then 
So we found the domain of definition of the function we needed, it is the set of all real numbers except zero.
Answer:
(−∞, 0)∪(0, +∞) .
Example.
What is the domain of a function 
?
Solution.
This function is complex, it can be considered as y=f 1 (f 2 (x)), where f 1 is a power function with exponent, and f 2 is the arcsine function, and we need to find its domain of definition.
Let's see what we know: D(f 1)=(0, +∞) and D(f 2)=[−1, 1] . It remains to find the intersection of sets of values x such that x∈D(f 2) and f 2 (x)∈D(f 1) : 
To arcsinx>0, remember the properties of the arcsine function. The arcsine increases throughout the entire domain of definition [−1, 1] and goes to zero at x=0, therefore, arcsinx>0 for any x from the interval (0, 1] .
Let's return to the system: 
Thus, the required domain of definition of the function is the half-interval (0, 1].
Answer:
(0, 1] .
Now let's move on to complex functions general view y=f 1 (f 2 (…f n (x)))) . The domain of definition of the function f in this case is found as 
.
Example.
Find the domain of a function 
.
Solution.
Given complex function can be written as y=f 1 (f 2 (f 3 (x))), where f 1 – sin, f 2 – fourth-degree root function, f 3 – log.
We know that D(f 1)=(−∞, +∞) , D(f 2)=- ∞; + ∞[ .
Example 1. Find the domain of a function y = 2 .
Solution. The domain of definition of the function is not indicated, which means that by virtue of the above definition, the natural domain of definition is meant. Expression f(x) = 2 defined for any real values x, hence, this function defined on the entire set R real numbers.
Therefore, in the drawing above, the number line is shaded all the way from minus infinity to plus infinity.
Root definition area n th degree
In the case when the function is given by the formula and n- natural number:
Example 2. Find the domain of a function .
Solution. As follows from the definition, a root of an even degree makes sense if the radical expression is non-negative, that is, if - 1 ≤ x≤ 1. Therefore, the domain of definition of this function is [- 1; 1] .
The shaded area of the number line in the drawing above is the domain of definition of this function.
Domain of power function
Domain of a power function with an integer exponent
If a- positive, then the domain of definition of the function is the set of all real numbers, that is ]- ∞; + ∞[ ;
If a- negative, then the domain of definition of the function is the set ]- ∞; 0[ ∪ ]0 ;+ ∞[ , that is, the entire number line except zero.
In the corresponding drawing above, the entire number line is shaded, and the point corresponding to zero is punched out (it is not included in the domain of definition of the function).
Example 3. Find the domain of a function .
Solution. First term whole degree x equals 3, and the degree of x in the second term can be represented as one - also an integer. Consequently, the domain of definition of this function is the entire number line, that is ]- ∞; + ∞[ .
Domain of a power function with a fractional exponent
In the case when the function is given by the formula:
if is positive, then the domain of definition of the function is the set 0; + ∞[ .
Example 4. Find the domain of a function .
Solution. Both terms in the function expression are power functions with positive fractional exponents. Consequently, the domain of definition of this function is the set - ∞; + ∞[ .
Domain of exponential and logarithmic functions
Domain of the exponential function
In the case when a function is given by a formula, the domain of definition of the function is the entire number line, that is ] - ∞; + ∞[ .
Domain of the logarithmic function
The logarithmic function is defined provided that its argument is positive, that is, its domain of definition is the set ]0; + ∞[ .
Find the domain of the function yourself and then look at the solution
Domain of trigonometric functions
Function Domain y= cos( x) - also many R real numbers.
Function Domain y= tg( x) - a bunch of R
real numbers other than numbers 
.
Function Domain y= ctg( x) - a bunch of R real numbers, except numbers.
Example 8. Find the domain of a function .
Solution. External function - decimal logarithm and the domain of its definition is subject to the conditions of the domain of definition of the logarithmic function in general. That is, her argument must be positive. The argument here is the sine of "x". Turning an imaginary compass around a circle, we see that the condition sin x> 0 is violated with "x" equal to zero, "pi", two, multiplied by "pi" and in general equal to the product pi and any even or odd integer.
Thus, the domain of definition of this function is given by the expression
,
Where k- an integer.
Domain of definition of inverse trigonometric functions
Function Domain y= arcsin( x) - set [-1; 1] .
Function Domain y= arccos( x) - also the set [-1; 1] .
Function Domain y= arctan( x) - a bunch of R real numbers.
Function Domain y= arcctg( x) - also many R real numbers.
Example 9. Find the domain of a function .
Solution. Let's solve the inequality:
Thus, we obtain the domain of definition of this function - the segment [- 4; 4] .
Example 10. Find the domain of a function
.
Solution. Let's solve two inequalities:
Solution to the first inequality:
Solution to the second inequality:
Thus, we obtain the domain of definition of this function - the segment.
Fraction scope
If a function is given by a fractional expression in which the variable is in the denominator of the fraction, then the domain of definition of the function is the set R real numbers, except these x, at which the denominator of the fraction becomes zero.
Example 11. Find the domain of a function .
Solution. By solving the equality of the denominator of the fraction to zero, we find the domain of definition of this function - the set ]- ∞; - 2[ ∪ ]- 2 ;+ ∞[ .
Fractional equations. ODZ.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
We continue to master the equations. We already know how to work with linear and quadratic equations. The last view left - fractional equations. Or they are also called much more respectably - fractional rational equations . It is the same.
Fractional equations.
As the name implies, these equations necessarily contain fractions. But not just fractions, but fractions that have unknown in denominator. At least in one. For example:
Let me remind you that if the denominators are only numbers, these are linear equations.
How to decide fractional equations? First of all, get rid of fractions! After this, the equation most often turns into linear or quadratic. And then we know what to do... In some cases it can turn into an identity, such as 5=5 or an incorrect expression, such as 7=2. But this rarely happens. I will mention this below.
But how to get rid of fractions!? Very simple. Applying the same identical transformations.
We need to multiply the entire equation by the same expression. So that all denominators are reduced! Everything will immediately become easier. Let me explain with an example. Let us need to solve the equation:
As taught in junior classes? We move everything to one side, bring it to a common denominator, etc. Forget how horrible dream! This is what you need to do when you add or subtract. fractional expressions. Or you work with inequalities. And in equations, we immediately multiply both sides by an expression that will give us the opportunity to reduce all denominators (i.e., in essence, by common denominator). And what is this expression?
On the left side, reducing the denominator requires multiplying by x+2. And on the right, multiplication by 2 is required. This means that the equation must be multiplied by 2(x+2). Multiply:
This ordinary multiplication fractions, but I’ll write it down in detail:
Please note that I am not opening the bracket yet (x + 2)! So, in its entirety, I write it:
On the left side it contracts entirely (x+2), and on the right 2. Which is what was required! After reduction we get linear the equation:
And everyone can solve this equation! x = 2.
Let's solve another example, a little more complicated:
If we remember that 3 = 3/1, and 2x = 2x/ 1, we can write:
And again we get rid of what we don’t really like - fractions.
We see that to reduce the denominator with X, we need to multiply the fraction by (x – 2). And a few are not a hindrance to us. Well, let's multiply. All left side And all right side:
Parentheses again (x – 2) I'm not revealing. I work with the bracket as a whole as if it were one number! This must always be done, otherwise nothing will be reduced.
With a feeling of deep satisfaction we reduce (x – 2) and we get an equation without any fractions, with a ruler!
Now let’s open the brackets:
We bring similar ones, move everything to the left side and get:
But before that we will learn to solve other problems. On interest. That's a rake, by the way!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
In mathematics infinite set functions. And each has its own character.) To work with a wide variety of functions you need single an approach. Otherwise, what kind of mathematics is this?!) And there is such an approach!
When working with any function, we present it with standard set questions. And the first, the most important question- This domain of definition of a function. This area is sometimes called a set acceptable values argument, function specification area, etc.
What is the domain of a function? How to find it? These questions often seem complex and incomprehensible... Although, in fact, everything is extremely simple. You can see for yourself by reading this page. Go?)
Well, what can I say... Just respect.) Yes! The natural domain of a function (which is discussed here) matches with ODZ of expressions included in the function. Accordingly, they are searched according to the same rules.
Now let’s look at a not entirely natural domain of definition.)
Additional restrictions on the scope of a function.
Here we will talk about the restrictions that are imposed by the task. Those. the task contains some additional conditions, which were invented by the compiler. Or the restrictions emerge from the very method of defining the function.
As for the restrictions in the task, everything is simple. Usually, there is no need to look for anything, everything is already said in the task. Let me remind you that the restrictions written by the author of the task do not cancel fundamental limitations of mathematics. You just need to remember to take into account the conditions of the task.
For example, this task:
Find the domain of a function:
on the set of positive numbers.
We found the natural domain of definition of this function above. This area:
D(f)=( -∞ ; -1) ∪ (-1; 2] ∪ ∪
IN verbal way When specifying a function, you need to carefully read the condition and find restrictions on X there. Sometimes the eyes look for formulas, but the words whistle past the consciousness yes...) Example from the previous lesson:
The function is specified by the condition: each value of the natural argument x is associated with the sum of the digits that make up the value of x.
It should be noted here that we are talking only O natural values X. Then D(f) instantly recorded:
D(f): x ∈ N
As you can see, the scope of a function is not so complex concept. Finding this region comes down to examining the function, writing a system of inequalities, and solving this system. Of course, there are all kinds of systems, simple and complex. But...
I'll open it little secret. Sometimes a function for which you need to find the domain of definition looks simply intimidating. I want to turn pale and cry.) But as soon as I write down the system of inequalities... And, suddenly, the system turns out to be elementary! Moreover, often, the more terrible the function, the simpler the system...
Moral: the eyes fear, the head decides!)