All operations with the function can be performed only in the set where it is defined.
Therefore, when investigating a function and constructing its graph, the first role is played by finding the domain of definition.
In order to find the domain of definition of a function,It is necessary to find "dangerous zones", that is, such values of x for which the function does not exist and then exclude them from the set of real numbers. What is worth paying attention to?
If the function has the form y = g (x) / f (x), decideThe inequality f (x) ≠ -0, because the denominator of the fraction can not be zero. For example, y = (x + 2) / (x - 4), x - 4 ≠ -0. That is, the domain of definition is the set (-∞- 4) ∪- (4- + ∞-).
When defining a function, there is a rootEven degree, solve the inequality, where the value under the root will be greater than or equal to zero. A root of even degree can be taken only from a nonnegative number. For example, y = √- (x - 2), then x - 2≥-0. Then the domain of definition is the set [2- + ∞-).
If the function contains a logarithm, decideInequality, where the expression under the logarithm must be greater than zero, because the domain of the definition of the logarithm is only positive. For example, y = lg (x + 6), that is, x + 6 & gt-0 and the domain of definition is (-6- + ∞-).
It is worth paying attention if the function containsTangent or cotangent. The domain of the function tg (x) is all numbers except x = Π- / 2 + Π- * n, ctg (x) are all numbers except x = Π- * n, where n takes integer values. For example, y = tg (4 * x), that is 4 * x ≠ -Π- / 2 + Π- * n. Then the domain of definition (-∞- Π- / 8 + Π- * n / 4) ∪- (Π- / 8 + Π- * n / 4- + ∞-).
Remember that inverse trigonometric functions- arcsine and arc cosine are defined on the interval [-1-1], that is, if y = arcsin (f (x)) or y = arccos (f (x)), the double inequality -1≤-f (x) ≤-1. For example, y = arccos (x + 2), -1≤-x + 2≤-1. The domain of definition is the interval [-3 - -1].
Finally, if a combination of differentFunctions, the domain of definition is the intersection of the domains of definition of all these functions. For example, y = sin (2 * x) + x / √- (x + 2) + arcsin (x - 6) + lg (x - 6). First, find the domain of all terms. Sin (2 * x) is defined on the entire number line. For the function x / √- (x + 2), solve the inequality x + 2 & gt-0 and the domain of definition is (-2- + ∞-). The domain of the function arcsin (x - 6) is defined by the double inequality -1≤-x-6≤-1, that is, the interval [5-7] is obtained. For the logarithm, the inequality x - 6 & gt-0 holds, and this is the interval (6 - + ∞-). Thus, the domain of definition of the function is the set (-∞- + ∞-) ∩ - (- 2- + ∞-) ∩- [5- 7] ∩- (6- + ∞-), that is (6-7) .