How to determine d from f. Inverse trigonometric functions

Many problems lead us to search for a set of function values on a certain segment or throughout the entire domain of definition. Such tasks include various evaluations of expressions and solving inequalities.

In this article, we will define the range of values of a function, consider methods for finding it, and analyze in detail the solution of examples from simple to more complex. All material will be provided with graphic illustrations for clarity. So this article is a detailed answer to the question of how to find the range of a function.

Definition.

The set of values of the function y = f(x) on the interval X is the set of all values of a function that it takes when iterating over all .

Definition.

Function range y = f(x) is the set of all values of a function that it takes when iterating over all x from the domain of definition.

The range of the function is denoted as E(f) .

The range of a function and the set of values of a function are not the same thing. We will consider these concepts equivalent if the interval X when finding the set of values of the function y = f(x) coincides with the domain of definition of the function.

Also, do not confuse the range of the function with the variable x for the expression on the right side of the equality y=f(x) . The range of permissible values of the variable x for the expression f(x) is the domain of definition of the function y=f(x) .

The figure shows several examples.

Graphs of functions are shown with thick blue lines, thin red lines are asymptotes, red dots and lines on the Oy axis show the range of values of the corresponding function.

As you can see, the range of values of a function is obtained by projecting the graph of the function onto the y-axis. She could be the one singular(first case), a set of numbers (second case), a segment (third case), an interval (fourth case), an open ray (fifth case), a union (sixth case), etc.

So what do you need to do to find the range of values of a function?

Let's start from the very beginning simple case: we'll show you how to define a set of values continuous function y = f(x) on the segment .

It is known that a function continuous on an interval reaches its maximum and minimum values on it. Thus, the set of values of the original function on the segment will be the segment . Consequently, our task comes down to finding the largest and smallest values of the function on the segment.

For example, let's find the range of values of the arcsine function.

Example.

Specify the range of the function y = arcsinx .

Solution.

The area of definition of the arcsine is the segment [-1; 1] . Let's find the greatest and smallest value functions on this segment.

The derivative is positive for all x from the interval (-1; 1), that is, the arcsine function increases over the entire domain of definition. Consequently, it takes the smallest value at x = -1, and the largest at x = 1.

We have obtained the range of arcsine function .

Example.

Find the set of function values on the segment.

Solution.

Let's find the largest and smallest value of the function on a given segment.

Let us determine the extremum points belonging to the segment:

We calculate the values of the original function at the ends of the segment and at points :

Therefore, the set of values of a function on an interval is the interval .

Now we will show how to find the set of values of a continuous function y = f(x) in the intervals (a; b) , .

First, we determine the extremum points, extrema of the function, intervals of increase and decrease of the function on a given interval. Next, we calculate at the ends of the interval and (or) the limits at infinity (that is, we study the behavior of the function at the boundaries of the interval or at infinity). This information is enough to find the set of function values on such intervals.

Example.

Define the set of function values on the interval (-2; 2) .

Solution.

Let's find the extremum points of the function falling on the interval (-2; 2):

Dot x = 0 is a maximum point, since the derivative changes sign from plus to minus when passing through it, and the graph of the function goes from increasing to decreasing.

there is a corresponding maximum of the function.

Let's find out the behavior of the function as x tends to -2 on the right and as x tends to 2 on the left, that is, we find one-sided limits:

What we got: when the argument changes from -2 to zero, the function values increase from minus infinity to minus one-fourth (the maximum of the function at x = 0), when the argument changes from zero to 2, the function values decrease to minus infinity. Thus, the set of function values on the interval (-2; 2) is .

Example.

Specify the set of values of the tangent function y = tgx on the interval.

Solution.

The derivative of the tangent function on the interval is positive , which indicates an increase in function. Let's study the behavior of the function at the boundaries of the interval:

Thus, when the argument changes from to, the function values increase from minus infinity to plus infinity, that is, the set of tangent values on this interval is the set of all real numbers.

Example.

Find the range of a function natural logarithm y = lnx.

Solution.

The natural logarithm function is defined for positive values argument . On this interval the derivative is positive , this indicates an increase in the function on it. Let's find the one-sided limit of the function as the argument tends to zero on the right, and the limit as x tends to plus infinity:

We see that as x changes from zero to plus infinity, the values of the function increase from minus infinity to plus infinity. Therefore, the range of the natural logarithm function is the entire set of real numbers.

Example.

Solution.

This function is defined for all real values of x. Let us determine the extremum points, as well as the intervals of increase and decrease of the function.

Consequently, the function decreases at , increases at , x = 0 is the maximum point, the corresponding maximum of the function.

Let's look at the behavior of the function at infinity:

Thus, at infinity the values of the function asymptotically approach zero.

We found out that when the argument changes from minus infinity to zero (the maximum point), the function values increase from zero to nine (to the maximum of the function), and when x changes from zero to plus infinity, the function values decrease from nine to zero.

Look at the schematic drawing.

Now it is clearly visible that the range of values of the function is .

Finding the set of values of the function y = f(x) on intervals requires similar research. We will not dwell on these cases in detail now. We will meet them again in the examples below.

Let the domain of definition of the function y = f(x) be the union of several intervals. When finding the range of values of such a function, the sets of values on each interval are determined and their union is taken.

Example.

Find the range of the function.

Solution.

The denominator of our function should not go to zero, that is, .

First, let's find the set of function values on the open ray.

Derivative of a function is negative on this interval, that is, the function decreases on it.

We found that as the argument tends to minus infinity, the function values asymptotically approach unity. When x changes from minus infinity to two, the values of the function decrease from one to minus infinity, that is, on the interval under consideration, the function takes on a set of values. We do not include unity, since the values of the function do not reach it, but only asymptotically tend to it at minus infinity.

We proceed similarly for open beam.

On this interval the function also decreases.

The set of function values on this interval is the set .

Thus, the desired range of values of the function is the union of the sets and .

Graphic illustration.

Special attention should be paid to periodic functions. Range of values periodic functions coincides with the set of values on the interval corresponding to the period of this function.

Example.

Find the range of the sine function y = sinx.

Solution.

This function is periodic with a period of two pi. Let's take a segment and define the set of values on it.

The segment contains two extremum points and .

We calculate the values of the function at these points and on the boundaries of the segment, select the smallest and highest value:

Hence, .

Example.

Find the range of a function .

Solution.

We know that the arc cosine range is the segment from zero to pi, that is, or in another post. Function can be obtained from arccosx by shifting and stretching along the abscissa axis. Such transformations do not affect the range of values, therefore, . Function obtained from stretching three times along the Oy axis, that is, . And the last stage of transformation is a shift of four units down along the ordinate. This leads us to double inequality

Thus, the required range of values is .

Let us give the solution to another example, but without explanations (they are not required, since they are completely similar).

Example.

Define Function Range .

Solution.

Let us write the original function in the form . The range of values of the power function is the interval. That is, . Then

Hence, .

To complete the picture, we should talk about finding the range of values of a function that is not continuous on the domain of definition. In this case, we divide the domain of definition into intervals by break points, and find sets of values on each of them. By combining the resulting sets of values, we obtain the range of values of the original function. We recommend you remember

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(set of real numbers).

If a function is defined by a formula, then, in the absence of special reservations, the domain of its definition is considered to be the largest 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 get the real value of the 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 concerns generally accepted norms, for example, that one cannot divide 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 when "x" is 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 ;+ ∞[ .

Function y=f(x) is such a dependence of the variable y on the variable x, when each valid value of the variable x corresponds to a single value of the variable y.

Function definition domain D(f) is the set of all possible values of the variable x.

Function Range E(f) is the set of all admissible values of the variable y.

Graph of a function y=f(x) is a set of points on the plane whose coordinates satisfy a given functional dependence, that is, points of the form M (x; f(x)). The graph of a function is a certain line on a plane.

If b=0 , then the function will take the form y=kx and will be called direct proportionality.

D(f) : x \in R;\enspace E(f) : y \in R

The graph of a linear function is a straight line.

The slope k of the straight line y=kx+b is calculated using the following formula:

k= tan \alpha, where \alpha is the angle of inclination of the straight line to the positive direction of the Ox axis.

1) The function increases monotonically for k > 0.

For example: y=x+1

2) The function decreases monotonically as k< 0 .

For example: y=-x+1

3) If k=0, then giving b arbitrary values, we obtain a family of straight lines parallel to the Ox axis.

For example: y=-1

Inverse proportionality

Inverse proportionality called a function of the form y=\frac (k)(x), where k is a non-zero real number

D(f) : x \in \left \( R/x \neq 0 \right \); \: E(f) : y \in \left \(R/y \neq 0 \right \).

Function graph y=\frac (k)(x) is a hyperbole.

1) If k > 0, then the graph of the function will be located in the first and third quarters of the coordinate plane.

For example: y=\frac(1)(x)

2) If k< 0 , то график функции будет располагаться во второй и четвертой координатной плоскости.

For example: y=-\frac(1)(x)

Power function

Power function is a function of the form y=x^n, where n is a non-zero real number

1) If n=2, then y=x^2. D(f) : x \in R; \: E(f) : y \in; main period of the function T=2 \pi

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

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.

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 the function contains Square root, then the radical expression must be greater than or equal to zero. Remember that the square root of negative numbers cannot be taken. 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
1. 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.
2. 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 is 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
3. 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).
4. 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, ∞)
5. 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.

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

Next, the following examples are solved on the topic “Domain of definition of functions”.

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Example 14

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 of 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:

Since we can only extract the square root from a non-negative number, therefore, the function under the root is non-negative.

2x - 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)=)