How to find the scope of a function?
Very often, when performing tasks, a problem arises, how to find the scope of the function definition? Without this, there is no way to do it when constructing the graphs and further study the values of the function.
The concept of the domain of definition of a function
The domain of the function is the setvalues of the variable X for which the function f (X) has a meaning. More precisely, the value of the variable of the function X, for which f (X) can exist in reality, will be more precise. For example, it is suggested to consider the case when the function can not exist at all. The first case that we will consider when in the expression. In the variant, when there is a fraction, the denominator should not be equal to zero, for the simple reason that such fractional expressions simply do not exist, since they eventually result in a value of zero, and one of the golden rules of arithmetic - one can not divide by zero.
With zero sorted out, let's deal with the mostshot. To find the domain of the function definition, examples with the same fraction, and determine the value of the variable X, we need to equalize the fraction to zero, and solving this equation, we get 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, since when solving such a function, for any variant of the radicand, we get a positive answer, which will be removed from the domain of the function definition. What can not be said about the root of an odd degree, when we are satisfied only with a positively sub-root number.
Examples of solutions
Another example, when it is necessary to find the domain of definitiongiven function, given by the logarithm. Here it is quite simple, the domain of the definition of the logarithm is all positive numbers. And to find the values of the variable, we must solve the inequality for a given logarithm. Where the sub-rhythmical expression is negative. We must also take into account the inverse trigonometric functions, namely the arcsine and arc cosine, which are determined on the interval [-1: 1]. To do this, we must make sure that the meaning of the expression indicated by these functions falls into the known interval in advance, and all the rest is safely excluded from the values of the variable.
One example of how to find the scope of a definitionfunction if the function contains, for example, a compound fraction. Where, for example, the denominator will look like the root of the arcsine. In this case, it is necessary to select only those values of the variable for which the arcsine can exist, and already from them we remove the arcsine value which is equal to zero (since this is the denominator in this example), the next step excludes all negative values, for the simple reason that they do not suit the function of the sub-root value. All remaining values are required.
Suppose that our function has the form y = a / b, itsThe scope of the definition is all values except zero. The value of the number A can be completely arbitrary. For example, to find the domain for determining the data of the function y = 3 / 2x-1, we need to find those values of X for which the denominator of the given fraction does not vanish. To this end, we equate the denominator with zero and find a solution, after which y on c obtains a response equal to 0.5 (x: 2x-1 = 0; 2x = 1; x = 1; x = 0.5). Following from this, define the function, the value 0.5 should be omitted. In order to find the domain of the function definition, the solution must take into account that the given expression must be either positive or equal to zero.
It is necessary to find the domain of definition of the function= √3x-9, based on the above condition, we transform our expression into the form of the inequality 3x ≥ 9; x ≥ 3; 0, after the solution, which we arrive at the value that x is greater than or equal to 3, and exclude all these values from the function area. When determining the domain of definition of the function of the radicand with an odd exponent, it must be taken into account that in this case the value of X can be , if the root expression is not fractional, and X is not in the denominator. example: y = ³√2x-5, you can simply indicate that the variable X can be any absolute number. In how to find the scope of the function definition, in no case should we forget that a given number under the logarithm must be positive.
Example: It is necessary to find the domain of definition of the data of the function y = log2 (4x - 1). Taking into account the above condition, the determination of the value of this function should be calculated as follows, 4x - 1> 0; this implies 4x> 1; x> 0.25. And the scope of the given function will be equal to all values greater than 0.25.
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