We found out that there is X- a set on which the formula that defines the function makes sense. IN mathematical analysis this set is often denoted as D (domain of a function ). In turn, many Y denoted as E (function range ) and wherein D And E called subsets R(sets real numbers).
If a function is given by a formula, then, in the absence of special reservations, the scope of its definition is considered greatest set, on which this formula makes sense, that is, the largest set of argument values that leads to real values of the function . In other words, the set of argument values on which the “function works”.
For common understanding The example does not yet have a formula. The function is specified as pairs of relations:
{(2, 1), (4, 2), (6, -6), (5, -1), (7, 10)} .
Find the domain of definition of these functions.
Answer. The first element of the pair is a variable x. Since the function specification also contains the second elements of the pairs - the values of the variable y, then the function makes sense only for those values of x that correspond to certain value game. That is, we take all the X’s of these pairs in ascending order and obtain from them the domain of definition of the function:
{2, 4, 5, 6, 7} .
The same logic works if the function is given by a formula. Only the second elements in pairs (that is, the values of the i) are obtained by substituting certain x values into the formula. However, to find the domain of a function, we do not need to go through all the pairs of X's and Y's.
Example 0. How to find the domain of definition of the function i is equal to the square root of x minus five (radical expression x minus five) ()? You just need to solve the inequality
x - 5 ≥ 0 ,
since in order for us to receive real value game, the radical expression must be greater than or equal to zero. We get the solution: the domain of definition of the function is all values of x greater than or equal to five (or x belongs to the interval from five inclusive to plus infinity).
On the drawing above is a fragment of the number axis. On it, the region of definition of the considered function is shaded, while in the “plus” direction the hatching continues indefinitely along with the axis itself.
If you use computer programs, which produce some kind of answer based on the entered data, you may notice that for some values of the entered data the program displays an error message, that is, that with such data the answer cannot be calculated. This message is provided by the authors of the program if the expression for calculating the answer is quite complex or concerns some narrow subject area, or provided by the authors of the programming language, if it comes to generally accepted norms, for example, which cannot be divided by zero.
But in both cases, the answer (the value of some expression) cannot be calculated for the reason that the expression does not make sense for some data values.
An example (not quite mathematical yet): if the program displays the name of the month based on the month number in the year, then by entering “15” you will receive an error message.
Most often, the expression being calculated is just a function. Therefore they are not valid values data is not included domain of a function . And in hand calculations, it is just as important to represent the domain of a function. For example, you calculate a certain parameter of a certain product using a formula that is a function. For some values of the input argument, you will get nothing at the output.
Domain of definition of a constant
Constant (constant) defined for any real values x R real numbers. This can also be written like this: the domain of definition of this function is the entire number line ]- ∞; + ∞[ .
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 logarithmic function at all. 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 the function is given 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 ;+ ∞[ .
How ?
Examples of solutions
If something is missing somewhere, it means there is something somewhere
We continue to study the “Functions and Graphs” section, and the next station on our journey is. Active discussion this concept began in the article about sets and continued in the first lesson about function graphs, where I looked at elementary functions, and, in particular, their domains of definition. Therefore, I recommend that dummies start with the basics of the topic, since I will not dwell on some basic points again.
The reader is assumed to know the domain of definition following functions: linear, quadratic, cubic function, polynomials, exponential, sine, cosine. They are defined on (the set of all real numbers). For tangents, arcsines, so be it, I forgive you =) - rarer graphs are not immediately remembered.
The scope of definition seems to be a simple thing, and a logical question arises: what will the article be about? In this lesson I will look at common problems of finding the domain of a function. Moreover, we will repeat inequalities with one variable, the solution skills of which will be required in other tasks higher mathematics. The material, by the way, is all school material, so it will be useful not only for students, but also for students. The information, of course, does not pretend to be encyclopedic, but here are not far-fetched “dead” examples, but roasted chestnuts, which are taken from real practical works.
Let's start with a quick dive into the topic. Briefly about the main thing: we are talking about a function of one variable. Its domain of definition is many meanings of "x", for which exist meanings of "players". Let's consider conditional example:
The domain of definition of this function is a union of intervals:
(for those who have forgotten: - unification icon). In other words, if you take any value of “x” from the interval , or from , or from , then for each such “x” there will be a value “y”.
Roughly speaking, where the domain of definition is, there is a graph of the function. But the half-interval and the “tse” point are not included in the definition area and there is no graph there.
How to find the domain of a function? Many people remember the children's rhyme: “rock, scissors, paper,” and in in this case it can be safely paraphrased: “root, fraction and logarithm.” Thus, if you life path encounters a fraction, root or logarithm, you should immediately be very, very wary! Tangent, cotangent, arcsine, arccosine are much less common, and we will also talk about them. But first, sketches from the life of ants:
Domain of a function that contains a fraction
Suppose we are given a function containing some fraction . As you know, you cannot divide by zero: , so those “X” values that turn the denominator to zero are not included in the scope of this function.
I won’t dwell on the most simple functions like 
etc., since everyone perfectly sees points that are not included in their domain of definition. Let's look at more meaningful fractions:
Example 1
Find the domain of a function
Solution: There is nothing special in the numerator, but the denominator must be non-zero. Let's set it equal to zero and try to find the “bad” points:
The resulting equation has two roots: 
. Data values are not in the scope of the function. Indeed, substitute or into the function and you will see that the denominator goes to zero.
Answer: domain: 
The entry reads like this: “the domain of definition is all real numbers with the exception of the set consisting of values 
" Let me remind you that the backslash symbol in mathematics denotes logical subtraction, and curly brackets denote set. The answer can be equivalently written as a union of three intervals:
Whoever likes it.
At points 
function tolerates endless breaks, and straight lines, given by equations 
are vertical asymptotes for the graph of this function. However, this is a slightly different topic, and further I will not focus much attention on this.
Example 2
Find the domain of a function
The task is essentially oral and many of you will almost immediately find the area of definition. The answer is at the end of the lesson.
Will a fraction always be “bad”? No. For example, a function is defined on the entire number line. No matter what value of “x” we take, the denominator will not go to zero, moreover, it will always be positive: . Thus, the scope of this function is: .
All functions like 
defined and continuous on .
The situation is a little more complicated when the denominator is occupied quadratic trinomial:
Example 3
Find the domain of a function 
Solution: Let's try to find the points at which the denominator goes to zero. For this we will decide quadratic equation:
The discriminant turned out to be negative, which means real roots no, and our function is defined on the entire number line.
Answer: domain:
Example 4
Find the domain of a function 
This is an example for independent decision. The solution and answer are at the end of the lesson. I advise you not to be lazy with simple problems, since misunderstandings will accumulate with further examples.
Domain of a function with a root
The square root function is defined only for those values of "x" when radical expression is non-negative: . If the root is located in the denominator , then the condition is obviously tightened: . Similar calculations are valid for any root of positive even degree: 
, however, the root is already of the 4th degree in function studies I don't remember.
Example 5
Find the domain of a function 
Solution: the radical expression must be non-negative:
Before continuing with the solution, let me remind you of the basic rules for working with inequalities, known from school.
Please note Special attention! Now we are considering inequalities with one variable- that is, for us there is only one dimension along the axis. Please do not confuse with inequalities of two variables, where geometrically all coordinate plane. However, there are also pleasant coincidences! So, for inequality the following transformations are equivalent:
1) The terms can be transferred from part to part by changing their (the terms) signs.
2) Both sides of the inequality can be multiplied by a positive number.
3) If both sides of the inequality are multiplied by negative number, then you need to change sign of inequality itself. For example, if there was “more”, then it will become “less”; if it was “less than or equal”, then it will become “greater than or equal”.
In the inequality, we move the “three” to the right side with a change of sign (rule No. 1):
Let's multiply both sides of the inequality by –1 (rule No. 3):
Let's multiply both sides of the inequality by (rule No. 2):
Answer: domain: 
The answer can also be written in an equivalent phrase: “the function is defined at .”
Geometrically, the definition area is depicted by shading the corresponding intervals on the abscissa axis. In this case:
I remind you once again geometric meaning domain of definition – graph of a function 
exists only in the shaded area and is absent at .
In most cases, a purely analytical determination of the domain of definition is suitable, but when the function is very complicated, you should draw an axis and make notes.
Example 6
Find the domain of a function
This is an example for you to solve on your own.
When there is a square binomial or trinomial under the square root, the situation becomes a little more complicated, and now we will analyze in detail the solution technique:
Example 7
Find the domain of a function 
Solution: the radical expression must be strictly positive, that is, we need to solve the inequality. At the first step, we try to factor the quadratic trinomial: 
The discriminant is positive, we are looking for roots: 
So the parabola 
intersects the abscissa axis at two points, which means that part of the parabola is located below the axis (inequality), and part of the parabola is located above the axis (the inequality we need).
Since the coefficient is , the branches of the parabola point upward. From the above it follows that the inequality is satisfied on the intervals (the branches of the parabola go upward to infinity), and the vertex of the parabola is located on the interval below the x-axis, which corresponds to the inequality:
! Note:
If you don't fully understand the explanations, please draw the second axis and the entire parabola! It is advisable to return to the article and manual Hot formulas for school mathematics course.
Please note that the points themselves are removed (not included in the solution), since our inequality is strict.
Answer: domain:
In general, many inequalities (including the one considered) are solved by the universal interval method, known again from school curriculum. But in the cases of square binomials and trinomials, in my opinion, it is much more convenient and faster to analyze the location of the parabola relative to the axis. And we will analyze the main method - the interval method - in detail in the article. Function zeros. Constancy intervals.
Example 8
Find the domain of a function
This is an example for you to solve on your own. The sample comments in detail on the logic of the reasoning + the second method of solution and one more important transformation inequality, without knowledge of which a student will limp on one leg..., ...hmm... regarding the leg, perhaps, I got excited, rather, on one toe. Thumb.
Can a square root function be defined on the entire number line? Certainly. All familiar faces: . Or a similar sum with an exponent: . Indeed, for any values of “x” and “ka”: , therefore also and .
But less obvious example: 
. Here the discriminant is negative (the parabola does not intersect the x-axis), while the branches of the parabola are directed upward, hence the domain of definition: .
The opposite question: can the domain of definition of a function be empty? Yes, and a primitive example immediately suggests itself 
, where the radical expression is negative for any value of “x”, and the domain of definition: (icon empty set). Such a function is not defined at all (of course, the graph is also illusory).
With odd roots 
etc. everything is much better - here radical expression can be negative. For example, a function is defined on the entire number line. However, the function has a single point that is still not included in the domain of definition, since the denominator is set to zero. For the same reason for the function 
points are excluded.
Domain of a function with a logarithm
The third common function is the logarithm. As a sample I will draw natural logarithm, which occurs in approximately 99 examples out of 100. If a certain function contains a logarithm, then its domain of definition should include only those values of “x” that satisfy the inequality. If the logarithm is in the denominator: , then additionally a condition is imposed (since ).
Example 9
Find the domain of a function
Solution: in accordance with the above, we will compose and solve the system: 
Graphic solution for Dummies:
Answer: domain:
I'll stop at one more technical point– I don’t have the scale indicated and the divisions along the axis are not marked. The question arises: how to make such drawings in a notebook on checkered paper? Should the distance between points be measured by cells strictly according to scale? It is more canonical and stricter, of course, to scale, but a schematic drawing that fundamentally reflects the situation is also quite acceptable.
Example 10
Find the domain of a function 
To solve the problem, you can use the method of the previous paragraph - analyze how the parabola is located relative to the x-axis. The answer is at the end of the lesson.
As you can see, in the realm of logarithms everything is very similar to the situation with square roots: the function 
(square trinomial from Example No. 7) is defined on the intervals, and the function 
(square binomial from Example No. 6) on the interval . It’s awkward to even say, type functions are defined on the entire number line.
Helpful information
: interesting typical function, it is defined on the entire number line except the point. According to the property of the logarithm, the “two” can be multiplied outside the logarithm, but in order for the function not to change, the “x” must be enclosed under the modulus sign: 
. Here's another one for you" practical use» module =). This is what you need to do in most cases when you demolish even degree, for example: 
. If the base of the degree is obviously positive, for example, then there is no need for the modulus sign and it is enough to use parentheses: .
To avoid repetition, let's complicate the task:
Example 11
Find the domain of a function 
Solution: in this function we have both the root and the logarithm.
The radical expression must be non-negative: , and the expression under the logarithm sign must be strictly positive: . Thus, it is necessary to solve the system:
Many of you know very well or intuitively guess that the system solution must satisfy to each condition.
Examining the location of the parabola relative to the axis, we come to the conclusion that the inequality is satisfied by the interval (blue shading):
The inequality obviously corresponds to the “red” half-interval.
Since both conditions must be met simultaneously, then the solution to the system is the intersection of these intervals. " Common interests» are met on the half-interval.
Answer: domain:
The typical inequality, as demonstrated in Example No. 8, is not difficult to resolve analytically.
The found domain will not change for “similar functions”, e.g. 
or 
. You can also add some continuous functions, for example: , or like this: 
, or even like this: . As they say, the root and the logarithm are stubborn things. The only thing is that if one of the functions is “reset” to the denominator, then the domain of definition will change (although in general case this is not always true). Well, in the matan theory about this verbal... oh... there are theorems.
Example 12
Find the domain of a function 
This is an example for you to solve on your own. Using a drawing is quite appropriate, since the function is not the simplest.
A couple more examples to reinforce the material:
Example 13
Find the domain of a function
Solution: let’s compose and solve the system: 
All actions have already been discussed throughout the article. Let us depict the interval corresponding to the inequality on the number line and, according to the second condition, eliminate two points:
The meaning turned out to be completely irrelevant.
Answer: domain
A little math pun on a variation of the 13th example:
Example 14
Find the domain of a function 
This is an example for you to solve on your own. Those who missed it are out of luck ;-)
The final section of the lesson is devoted to more rare, but also “working” functions:
Function Definition Areas
with tangents, cotangents, arcsines, arccosines
If some function includes , then from its domain of definition excluded points 
, Where Z– a set of integers. In particular, as noted in the article Graphs and properties of elementary functions, the function has the following values:
That is, the domain of definition of the tangent: 
.
Let's not kill too much:
Example 15
Find the domain of a function
Solution: in this case, the following points will not be included in the domain of definition:
Let's throw the "two" of the left side into the denominator of the right side:
As a result 
:
Answer: domain: 
.
In principle, the answer can also be written as a union infinite number intervals, but the design will be very cumbersome:
The analytical solution is completely consistent with geometric transformation of the graph: if the argument of a function is multiplied by 2, then its graph will shrink to the axis twice. Notice how the function's period has been halved, and break points doubled in frequency. Tachycardia.
Similar story with cotangent. If some function includes , then the points are excluded from its domain of definition. In particular, for the automatic burst function we shoot the following values:
In other words:
In mathematics there is a fairly small number elementary functions, the scope of which is limited. All other "complex" functions are just combinations and combinations of them.
1. Fractional function - restriction on the denominator.
2. Root of even degree - restriction on radical expression.
3. Logarithms - restrictions on the base of the logarithm and sublogarithmic expression.
3. Trigonometric tg(x) and ctg(x) - restriction on the argument.
For tangent:
4. Inverse trigonometric functions.
| arcsine | arc cosine | Arctangent, Arctangent |
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Next, the following examples are solved on the topic “Domain of definition of functions”.
| Example 1 | Example 2 |
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| Example 3 | Example 4 |
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| Example 5 | Example 6 |
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| Example 7 | Example 8 |
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| Example 9 | Example 10 |
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| Example 11 | Example 12 |
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| Example 13 | Example 14 |
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| Example 15 | Example 16 |
An example of finding the domain of definition of function No. 1
Finding the domain of definition of any linear function, i.e. functions of the first degree:
y = 2x + 3 - the equation defines a straight line on a plane.
Let's look carefully at the function and think about what numerical values we can substitute into the equation instead of the variable x?
Let's try to substitute the value x=0
Since y = 2 0 + 3 = 3 - we got numeric value, therefore the function exists for the given value of the variable x=0.
Let's try to substitute the value x=10
since y = 2·10 + 3 = 23 - the function exists for the given value of the variable x=10.
Let's try to substitute the value x=-10
since y = 2·(-10) + 3 = -17 - the function exists for the given value of the variable x = -10.
The equation defines a straight line on a plane, and a straight line has neither beginning nor end, therefore it exists for any values of x.
Note that no matter what numerical values we substitute into a given function instead of x, we will always get the numerical value of the variable y.
Therefore, the function exists for any value x ∈ R, or we write it like this: D(f) = R
Forms of writing the answer: D(f)=R or D(f)=(-∞:+∞)or x∈R or x∈(-∞:+∞)
Let's conclude:
For any function of the form y = ax + b, the domain of definition is the set of real numbers.
An example of finding the domain of definition of function No. 2
A function of the form:
y = 10/(x + 5) - hyperbola equation
When dealing with a fractional function, remember that you cannot divide by zero. Therefore the function will exist for all values of x that are not
set the denominator to zero. Let's try to substitute some arbitrary values X.
At x = 0 we have y = 10/(0 + 5) = 2 - the function exists.
For x = 10 we have y = 10/(10 + 5) = 10/15 = 2/3- the function exists.
At x = -5 we have y = 10/(-5 + 5) = 10/0 - the function does not exist at this point.
Those. If given function fractional, then it is necessary to equate the denominator to zero and find a point at which the function does not exist.
In our case:
x + 5 = 0 → x = -5 - at this point the given function does not exist.
x + 5 ≠ 0 → x ≠ -5
For clarity, let's depict it graphically:
On the graph we also see that the hyperbola comes as close as possible to the straight line x = -5, but does not reach the value -5 itself.
We see that the given function exists at all points of the real axis, except for the point x = -5
Response recording forms: D(f)=R\(-5) or D(f)=(-∞;-5) ∪ (-5;+∞) or x ∈ R\(-5) or x ∈ (-∞;-5) ∪ (-5;+∞)
If the given function is fractional, then the presence of a denominator imposes the condition that the denominator is not equal to zero.
An example of finding the domain of definition of function No. 3
Let's consider an example of finding the domain of definition of a function with a root of even degree:
Because Square root we can only extract from non-negative number, therefore, the function under the root is non-negative.
2х - 8 ≥ 0
Let's solve a simple inequality:
2x - 8 ≥ 0 → 2x ≥ 8 → x ≥ 4
The specified function exists only for the found values of x ≥ 4 or D(f)= ∪ ∪
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!)
Each function has two variables - an independent variable and a dependent variable, the values of which depend on the values of the independent variable. For example, in the function y = f(x) = 2x + y The independent variable is "x" and the dependent variable is "y" (in other words, "y" is a function of "x"). The valid values of the independent variable "x" are called the domain of the function, and the valid values of the dependent variable "y" are called the domain of the function.
Steps
Part 1
Finding the Domain of a Function-
Select the appropriate entry for the function's scope. The scope of definition is written in square and/or parentheses. Square bracket applies when the value is within the scope of the function; if the value is not within the scope of the definition, a parenthesis is used. If a function has several non-adjacent domains, a “U” symbol is placed between them.
- For example, the scope of [-2,10)U(10,2] includes the values -2 and 2, but does not include the value 10.
-
Plot a graph quadratic function. The graph of such a function is a parabola, the branches of which are directed either up or down. Since the parabola increases or decreases along the entire X-axis, the domain of definition of the quadratic function is all real numbers. In other words, the domain of such a function is the set R (R stands for all real numbers).
- To better understand the concept of a function, select any value of “x”, substitute it into the function and find the value of “y”. A pair of values “x” and “y” represent a point with coordinates (x,y) that lies on the graph of the function.
- Plot this point on the coordinate plane and do the same process with a different x value.
- By plotting several points on the coordinate plane, you get general idea about the form of the graph of a function.
-
If the function contains a fraction, set its denominator to zero. Remember that you cannot divide by zero. Therefore, by setting the denominator to zero, you will find values of "x" that are not within the domain of the function.
- For example, find the domain of the function f(x) = (x + 1) / (x - 1) .
- Here the denominator is: (x - 1).
- Equate the denominator to zero and find “x”: x - 1 = 0; x = 1.
- Write down the domain of definition of the function. The domain of definition does not include 1, that is, it includes all real numbers except 1. Thus, the domain of definition of the function is: (-∞,1) U (1,∞).
- The notation (-∞,1) U (1,∞) reads like this: the set of all real numbers except 1. The infinity symbol ∞ means all real numbers. In our example, all real numbers that are greater than 1 and less than 1 are included in the domain.
-
If a function contains a square root, then the radical expression must be greater than or equal to zero. Remember that the square root of negative numbers not extracted. Therefore, any value of “x” at which the radical expression becomes negative must be excluded from the domain of definition of the function.
- For example, find the domain of the function f(x) = √(x + 3).
- Radical expression: (x + 3).
- The radical expression must be greater than or equal to zero: (x + 3) ≥ 0.
- Find "x": x ≥ -3.
- The domain of this function includes the set of all real numbers that are greater than or equal to -3. Thus, the domain of definition is [-3,∞).
Part 2
Finding the range of a quadratic function-
Make sure you are given a quadratic function. The quadratic function has the form: ax 2 + bx + c: f(x) = 2x 2 + 3x + 4. The graph of such a function is a parabola, the branches of which are directed either up or down. Exist various methods finding the range of values of a quadratic function.
- The easiest way to find the range of a function containing a root or fraction is to graph the function using a graphing calculator.
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Find the x coordinate of the vertex of the function graph. For a quadratic function, find the x coordinate of the vertex of the parabola. Remember that the quadratic function is: ax 2 + bx + c. To calculate the x coordinate, use the following equation: x = -b/2a. This equation is the derivative of the fundamental quadratic function and describes the tangent, slope which equal to zero(the tangent to the vertex of the parabola is parallel to the X axis).
- For example, find the range of the function 3x 2 + 6x -2.
- Calculate the x coordinate of the vertex of the parabola: x = -b/2a = -6/(2*3) = -1
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Find the y-coordinate of the vertex of the function graph. To do this, substitute the found “x” coordinate into the function. Searched coordinate"y" represents the limit value of the function range.
- Calculate the y coordinate: y = 3x 2 + 6x – 2 = 3(-1) 2 + 6(-1) -2 = -5
- The coordinates of the vertex of the parabola of this function are (-1,-5).
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Determine the direction of the parabola by plugging in at least one x value into the function. Choose any other x value and plug it into the function to calculate the corresponding y value. If the found “y” value is greater than the “y” coordinate of the vertex of the parabola, then the parabola is directed upward. If the found “y” value is less than the “y” coordinate of the vertex of the parabola, then the parabola is directed downward.
- Substitute into the function x = -2: y = 3x 2 + 6x – 2 = y = 3(-2) 2 + 6(-2) – 2 = 12 -12 -2 = -2.
- Coordinates of a point lying on the parabola: (-2,-2).
- The found coordinates indicate that the branches of the parabola are directed upward. Thus, the range of the function includes all values of "y" that are greater than or equal to -5.
- Range of values of this function: [-5, ∞)
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The domain of a function is written similarly to the domain of a function. The square bracket is used when the value is within the range of the function; if the value is not in the range, a parenthesis is used. If a function has several non-adjacent ranges of values, a “U” symbol is placed between them.
- For example, the range [-2,10)U(10,2] includes the values -2 and 2, but does not include the value 10.
- With the infinity symbol ∞, parentheses are always used.
Determine the type of function given to you. The range of values of the function is all valid “x” values (laid along the horizontal axis), which correspond to valid “y” values. The function can be quadratic or contain fractions or roots. To find the domain of a function, you first need to determine the type of the function.