We learned that there is X- a set on which the formula for which the function is given makes sense. In mathematical analysis, this set is often denoted as D (function domain ). In turn, many Y denoted as E (function range ) and wherein D and E called subsets R(sets of real numbers).
If a function is given by a formula, then, in the absence of special reservations, the domain of its definition is the largest set on which this formula makes sense, that is, the largest set of argument values that leads to the actual values of the function ... In other words, the set of argument values on which the "function works".
For a general understanding, the example is still without a formula. The function is specified as pairs of relations:
{(2, 1), (4, 2), (6, -6), (5, -1), (7, 10)} .
Find the domain of this function.
Answer. The first element of the pairs is a variable x... Since in the function definition, the second elements of the pairs are also given - the values of the variable y, then the function is meaningful only for those x values that correspond to a certain value of the game. That is, we take all Xs of these pairs in ascending order and get from them the domain of the function:
{2, 4, 5, 6, 7} .
The same logic works if the function is specified by a formula. Only the second elements in pairs (that is, the values of the game) are obtained by substituting certain x values into the formula. However, in order to find the domain of the function, we do not need to iterate over all pairs of Xs and Games.
Example 0. How to find the domain of definition of the function y 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 the function - all x values are greater than or equal to five (or x belongs to the interval from five inclusive to plus infinity).
In the drawing above, there is a fragment of the numerical axis. On it, the domain of definition of the function under consideration is shaded, while in the "plus" direction the shading continues indefinitely along with the axis itself.
If you use computer programs that, based on the entered data, give some kind of answer, you may notice that for some values of the entered data, the program gives an error message, that is, that with such data the answer cannot be calculated. Such a message is provided by the authors of the program, if the expression for calculating the answer is rather complicated or concerns some narrow subject area, or is 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 by the number of the month in the year, then entering "15" you will receive an error message.
More often than not, a calculated expression is actually a function. Therefore, such are not allowable values data not included in function domain ... And in freehand calculations, it is just as important to represent the domain of the function. For example, you calculate some parameter of some product using a formula that is a function. For some values of the argument at the input, you get nothing at the output.
Constant domain
Constant (constant) defined for any real values x R real numbers. It can be written like this: the domain 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, we mean the natural domain of definition. Expression f(x) = 2 is defined for any real values x, therefore, this function is defined on the entire set R real numbers.
Therefore, in the drawing above, the number line is shaded along the entire length from minus infinity to plus infinity.
Root scope 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, an even root makes sense if the radical expression is non-negative, that is, if - 1 ≤ x≤ 1. Therefore, the domain of this function is [- 1; 1] .
The shaded area of the number line in the drawing above is the definition area of this function.
Domain of power function
Domain of a power function with an integer exponent
if a- positive, then the domain of the function is the set of all real numbers, that is] - ∞; + ∞ [;
if a- negative, then the domain of the function is the set] - ∞; 0 [∪] 0; + ∞ [, that is, the entire number line except zero.
In the corresponding drawing from above, the entire number line is shaded, and the point corresponding to zero is punctured (it is not included in the domain of the function definition).
Example 3. Find the domain of a function .
Solution. The first term whole degree x, equal to 3, and the degree of x in the second term can be represented as one - also an integer. Therefore, the domain 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 - positive, then the domain 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 this function is the set - ∞; + ∞ [.
Domain of exponential and logarithmic functions
Domain of exponential function
In the case when a function is given by a formula, the domain of the function is the entire number line, that is] - ∞; + ∞ [.
Domain of the logarithmic function
The logarithmic function is defined under the condition that its argument is positive, that is, the domain of its definition is the set] 0; + ∞ [.
Find the scope of the function yourself and then see the solution
Domain of trigonometric functions
Function scope y= cos ( x) is also the set R real numbers.
Function scope y= tg ( x) - lots of R
real numbers other than numbers 
.
Function scope y= ctg ( x) - lots of R real numbers other than 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 x sine. Turning an imaginary compass in 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 the number "pi" and any even or odd integer.
Thus, the domain of this function is given by the expression
,
where k- an integer.
Domain of inverse trigonometric functions
Function scope y= arcsin ( x) - the set [-1; 1] .
Function scope y= arccos ( x) - also the set [-1; 1] .
Function scope y= arctg ( x) - lots of R real numbers.
Function scope y= arcctg ( x) is also the set 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 of the first inequality:
Solution of the second inequality:
Thus, we obtain the domain of definition of this function - a segment.
Fraction definition area
If the function is given by a fractional expression in which the variable is in the denominator of the fraction, then the domain of the function is the set R real numbers, except for such x at which the denominator of the fraction vanishes.
Example 11. Find the domain of a function .
Solution. Solving the equality to zero of the denominator of the fraction, we find the domain of definition of this function - the set] - ∞; - 2 [∪] - 2; + ∞ [.
Function y = f (x) is such a dependence of the y variable on the x variable, when each admissible value of the x variable corresponds to a single value of the y variable.
The scope of the function D (f) is the set of all admissible values of the variable x.
Function range E (f) is the set of all admissible values of the variable y.
Function graph y = f (x) is the set of points on the plane, the coordinates of which satisfy the 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
Linear function graph - straight line.
The slope k of the straight line y = kx + b is calculated using the following formula:
k = tg \ 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 proportion
Inverse proportion is called a function of the form y = \ frac (k) (x), where k is a nonzero 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 nonzero real number
1) If n = 2, then y = x ^ 2. D (f): x \ in R; \: E (f): y \ in; the main period of the function T = 2 \ pi
First, let's learn 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 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,…, fn, that is, the function f is given by the formula y = f 1 (x) + f 2 (x) +… + fn (x), then the domain of the function f is the intersection of the domains of definition of the functions f 1, f 2,…, fn. Let's write it as.
Let's agree to continue to use records like the last one, by which we mean written inside a curly brace, or the simultaneous fulfillment of any conditions. It is convenient and quite naturally echoes the meaning of the systems.
Example.
You are given a function y = x 7 + x + 5 + tgx, and you need to find its domain of definition.
Solution.
The function f is represented by the sum of four functions: f 1 is a power function with exponent 7, f 2 is a power function with exponent 1, f 3 is a constant function and f 4 is a tangent function.
Looking at the table of domains of definition of basic 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 for the numbers 
.
The domain of the function f is the intersection of the domains 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 numbers .
Answer:
the set of all real numbers except .
Moving on to finding domains of definition of the 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 the function f is the intersection of the domains of the functions f 1, f 2,…, f n. So, .
It is understandable, in the specified area all functions of the product 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 the function y = 3 · arctgx · lnx is the set of all real positive numbers.
Let us dwell separately on finding the domain of definition of the function given by the formula y = C · f (x), where C is some real number. It is easy to show that the domain of this function and the domain 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 the function f is D (f). Then the domain of the function y = C f (x) is 
, which was required 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 definition 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, complex function domain y = f 1 (f 2 (x)) is the intersection of two sets: the set of all x such that x∈D (f 2), and the set of all x such that f 2 (x) ∈D (f 1) ... That is, in our notation 
(this is essentially a system of inequalities).
Let's take a look at the solutions with a few examples. In the process, we will not describe in detail, since this is beyond the scope of this article.
Example.
Find the domain of the function y = lnx 2.
Solution.
The original function can be represented as y = f 1 (f 2 (x)), where f 1 is the logarithm with base e, and f 2 is a power function with exponent 2.
Turning to the well-known domains of definition of 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 need, it is the set of all real numbers except zero.
Answer:
(−∞, 0)∪(0, +∞) .
Example.
What is the scope of the 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 an 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): 
For arcsinx> 0, recall the properties of the arcsine function. The arcsine increases over the entire domain [−1, 1] and vanishes at x = 0, therefore, arcsinx> 0 for any x from the interval (0, 1].
Let's go back to the system: 
Thus, the required domain of definition of the function is a 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)))). In this case, the domain of the function f is found as 
.
Example.
Find the domain of a function 
.
Solution.
The given complex function can be written as y = f 1 (f 2 (f 3 (x))), where f 1 is sin, f 2 is the fourth root function, f 3 is lg.
We know that D (f 1) = (- ∞, + ∞), D (f 2) =.
Finally, if a combination of different functions is given, then the domain is the intersection of the domains of all these functions. For example, y = sin (2 * x) + x / √ (x + 2) + arcsin (x − 6) + log (x − 6). First, find the domain of definition of all terms. Sin (2 * x) is defined on the whole number line. For the function x / √ (x + 2), solve the inequality x + 2> 0 and the domain will be (-2; + ∞). The domain of the function arcsin (x − 6) is given by the double inequality -1≤x-6≤1, that is, a segment is obtained. For the logarithm, the inequality x − 6> 0 holds, and this is the interval (6; + ∞). Thus, the domain of the function will be the set (-∞; + ∞) ∩ (-2; + ∞) ∩∩ (6; + ∞), that is, (6; 7].
Related Videos
Sources:
- function domain with logarithm
A function is a concept that reflects the relationship between the elements of sets, or in other words, it is a "law" according to which each element of one set (called the domain of definition) is associated with some element of another set (called the domain of values).
Each function has two variables, the independent variable and the dependent variable, whose values 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 scope of the function. The scope is written in square and / or parentheses. A square bracket is used when a value is within the scope of a function; if the value is not in scope, a parenthesis is used. If the function has several non-contiguous domains of definition, the character "U" is placed between them.
- For example, the domain [-2,10) U (10,2] includes the values -2 and 2, but does not include the value 10.
-
Plot a 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 on the entire X-axis, the domain of the quadratic function is all real numbers. In other words, the domain of such a function is the set R (R denotes all real numbers).
- For a better understanding of the concept of a function, choose any value of "x", substitute it into the function and find the value "y". The pair of values "x" and "y" represent a point with coordinates (x, y), which lies on the graph of the function.
- Draw this point on the coordinate plane and follow the described process with a different "x" value.
- By plotting several points on the coordinate plane, you will get a general idea of the shape of the function graph.
-
If the function contains a fraction, set its denominator to zero. Remember that you cannot divide by zero. Therefore, equating the denominator to zero, you will find values for "x" that are not in the scope 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 scope of the function. The domain does not include 1, that is, it includes all real numbers except 1. Thus, the domain 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 greater than 1 and less than 1 are included in the scope.
-
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 not retrieved. Therefore, any value of "x", at which the radical expression becomes negative, must be excluded from the scope of the function.
- For example, find the domain of the function f (x) = √ (x + 3).
- The radical expression: (x + 3).
- The radical expression must be greater than or equal to zero: (x + 3) ≥ 0.
- Find "x": x ≥ -3.
- The scope of this function includes the set of all real numbers that are greater than or equal to -3. Thus, the domain is: [-3, ∞).
Part 2
Finding the Range of Values 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 whose branches are directed either up or down. Exists different methods finding the range of values of a quadratic function.
- The easiest way to find the range of a root or fraction function is to graph that function using a graphing calculator.
-
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 a derivative of the fundamental quadratic function and describes a tangent, the slope of which is 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
-
Find the y-coordinate of the vertex of the function graph. To do this, substitute the found coordinate "x" into the function. The sought-for coordinate "y" is the limiting value of the range of values of the function.
- 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).
-
Determine the direction of the parabola by substituting at least one x value into the function. Pick any other x value and plug it into the function to calculate the corresponding y value. If the found value "y" is greater than the coordinate "y" of the vertex of the parabola, then the parabola is directed upward. If the found value "y" is less than the coordinate "y" of the vertex of the parabola, then the parabola is directed downward.
- Substitute x = -2 in the function: y = 3x 2 + 6x - 2 = y = 3 (-2) 2 + 6 (-2) - 2 = 12 -12 -2 = -2.
- The coordinates of the point on the parabola are (-2, -2).
- The coordinates found indicate that the branches of the parabola are directed upward. Thus, the function's range includes all y values that are greater than or equal to -5.
- Range of values of this function: [-5, ∞)
-
The range of values of a function is written in the same way as the range of definition of a function. The square bracket is used when the value is in the range of the function; if the value is not in the range, a parenthesis is used. If a function has several non-contiguous ranges of values, the symbol "U" 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.
- Parentheses are always used with the infinity symbol ∞.
Determine the type of function given to you. The range of values of the function are all admissible values of "x" (plotted along the horizontal axis), which correspond to the admissible values of "y". 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.