The definition domain of a composite function refers to the set of parameters and values between two functions in a composite function that can adapt to each other.
A composite function is a new function formed by combining two or more functions with each other. Its definition domain depends on the relationship between the definition domains of the two functions. When determining the definition domain of a composite function, you need to consider whether the definition domains of the two functions overlap and the constraints of the function.
1.Define the basic concepts of the domain.
A defined domain is the range of values for the function's arguments (inputs), that is, the set of input values that make the function meaningful. For a single function, the defined domain can be a set of real numbers, a set of rational numbers, a set of integers, and so on. However, for composite functions, the determination of their definition domains needs to consider the relationship between the definition domains of the two functions.
2.Determine how to define the domain.
There are generally two approaches to determining the domain of a composite function: the direct method and the indirect method.
Direct method. The direct method is based on the compatibility between the defined domains of two functions. If the definition domains of two functions overlap, and the values of the functions are adapted to each other in the overlapping part, then the definition domain of the composite function will depend on the overlapping part.
For example, there are functions f(x) and g(x), and their definition domains are x and y, respectively. If there is an overlap between x and y, i.e., x y ≠ and for each x value in the overlap, the value of f(x) is within the defined domain of g(x), then the domain of the composite function h(x) = f(g(x)) is x y.
Indirect method. The indirect method is to deduce the range of values of the defined domain by determining the expression of the composite function. In general, the definition domain of the composite function will be determined by the definition domain of the inner function and the outer function, that is, the definition domain that needs to satisfy the inner function is included in the definition domain of the outer function.
There are functions f(x) and g(x), where the domain of f(x) is x and the domain of g(x) is y. If the domain y y of g(x) is within the range of the domain x of f(x) in the composite function h(x) = f(g(x)), then the domain of the composite function h(x) is y.
3.Defined domains for special cases.
When determining the domain of definition of a composite function, you also need to consider the constraints of the function. For example, when there is a fraction or root number in a function, you need to make the denominator or the number of open squares not equal to zero to ensure the integrity of the definition domain of the function.
There is the function f(x) = (g(x)), where g(x) is defined by x. Since the definition of the root function requires that the number of squares to be opened is greater than or equal to zero, the definition domain of the composite function h(x) = (g(x)) must satisfy g(x) 0.
4.The relationship between the function image and the defined domain.
For a given function image, the defined domain of the function can be inferred by looking at the image. The range of values of the independent variables covered by the function image is the definition domain of the function.
In summary, the domain defined by a composite function depends on the relationship between the domains defined by two functions. By direct or indirect method, the domain of the composite function can be determined. At the same time, it is also necessary to pay attention to the constraints of the function to ensure the integrity of the definition domain of the composite function.
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