• # How to find the domain of a function?

Very often when performing tasks there is a problem, how to find the domain of function? Without this, there is no way to do when plotting graphs and further studying the values ​​of the function.

## Understanding the scope of a function

The domain of a function is the set of values ​​of a variable of a function X for which the function f (X) makes sense. Or rather, to say the value of the variable of function X, for which f (X) can exist in reality. For example, it is proposed to consider the case when the function cannot exist at all. The first case, which we consider when in the expression. In the variant when a fraction takes place, the denominator must be non-zero, for the simple reason that such fractional expressions simply do not exist, since they ultimately lead to a value of zero, and one of the golden rules of arithmetic cannot be divided by zero.

Deal with zero, let's deal with the fraction itself. What to find the domain of the function, examples with the same fraction,and determine the value of the variable X, we need to equate the fraction to zero, and, having solved this equation, we obtain the value of the variable X, which will be excluded from the solution domain. The second example is when our function contains a root of even degree. Here we have complete freedom of action, because when solving such a function, we will get a positive answer for any variant of the radicand that will be further removed from the domain of the function. What can be said about the root of an odd degree, when we are satisfied with only a positively bred number.

## Examples of solutions

Another example is when you need to find the domain of data definition for a function defined by the logarithm. It's completely simple here, the domain of the logarithm is all positive numbers. And to find the values ​​of a variable, it is necessary to solve the inequality for a given logarithm. Where the sub-arithmetic expression will be negative. It is necessary to take into account the inverse trigonometric functions, namely, arc sine and arc cosine, which are defined on the interval [-1: 1]. To do this, we must make sure that the value of the expression indicated by these functions falls into a known interval in advance,and everything else can be safely excluded from the values ​​of the variable.

One example is how to find the domain of a function if the function contains, for example, a complex fraction. Where, for example, the denominator will look like a root of arcsine. In this case, it is necessary to select only those values ​​of the variable for which the arcsine can exist, and from them we remove the arcsine value which is zero (since it is in this example the denominator), the next step we exclude all negative values, for the simple reason that they are not satisfied with the condition of the function of the radicand. All remaining values ​​are required.

Suppose our function has the form y = a / b, its scope is all values ​​except for zero. The value of the number A can be anything. For example, to find the domain of data definition for the function y = 3 / 2x-1, we need to find those values ​​of X for which the denominator of this fraction will not be zero. To do this, we equate the denominator to zero and find the solution, after which y on c we get the answer equal to 0.5 (x: 2x - 1 = 0; 2x = 1; x = ½; x = 0.5) Following from this, from the region function definitions should exclude a value of 0.5.In order to find the domain of a function, the solution must take into account that this expression must either be positive or equal to zero.

We need to find the domain of definition of the function; examples y = √3x-9, based on the above condition, we transform our expression into the inequality type 3x ≥ 9; x ≥ 3; 0, after deciding which we will come to the value that x is greater than or equal to 3, and exclude all these values ​​from the function area. if the radicand is not a fractional, and X is not in the denominator. example: y = ³√2x-5, you can simply indicate that the variable X can be absolutely any real number. In how to find the domain of a function, in no case should we forget that the given number under the logarithm must be positive.

Example: It is necessary to find the domain of data definition for the function y = log2 (4x - 1). Given the above condition, finding the value of this function should be calculated as 4x - 1> 0; this implies 4x> 1; x> 0.25.And the domain of this function will be equal to all values ​​greater than 0.25.

Some sites offer to find the domain of function definition online and save time searching for solutions. Very convenient service, especially for students.

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