How to evaluate the value range of the composite function and how to find the definition domain of t

In the first part, we need to define the domain.

You can't be wrong about this.

The second part is to seek in layers.

For example, y=sin2x in the first layer, the definition range is r, so the function of the first level is y=2x in the value range.

is r and the domain of the function y=2x is the domain of the function y=sin(), so the domain of y=sin2x is the domain r of 2x

So the whole composite function.

The value range is -1,1

Start with the innermost function and evaluate the range, use this range as the definition domain of the outer function, and so on.

1. Find the defined domain of the function.

1. Use four operations and composite algorithms to reverse disassemble into simple functions;

2. Find the definition domain of each simple function;

3. Look at the conditions that the function needs to meet as a whole, for example, the denominator is not equal to zero, and the root number must be greater than or equal to zero;

4. Intersect all conditions.

2. Find the range of the function.

1. Find the inverse function first;

2. The defined domain of the inverse function is the value range;

3. For piecewise functions, the value range and inverse function should be found in segments, and the union set should be taken.

Finding the definition domain of a function should mainly consider the following points:

When it is an integer. or odd roots, the range of r;

When it is an even radical, the number of squares to be opened is not less than 0 (i.e., 0);

When it is a fraction. , the denominator is not 0; When the denominator is an even radical, the number of squares to be opened is greater than 0;

When exponential, the base is not 0 for the exponential power of zero or negative integer exponential power (e.g., medium).

When it is formed by combining some basic functions through four operations, its definition domain should be the set of values of independent variables that make each part meaningful, that is, find the intersection of the set of definition domains of each part.

The definition domain of a piecewise function is the union of the set of values of the independent variables on each segment.

Functions built by practical problems should consider not only the requirements of the arguments for the analytic expression, but also the requirements of the arguments for the practical meaning.

For functions with parameter letters, the values of the letters should be classified and discussed when finding the definition domain, and it should be noted that the definition domain of the function is a non-empty set.

Logarithmic functions. The true number of must be greater than zero, and the base number must be greater than zero and not equal to 1.

Trigonometric function. The cutting function in should be aware of the constraints of the diagonal variable.

First, find out the definition domain of the function, find which two or more functions are compounded by the function, the common one is the compound of two functions, determine the monotonicity according to the principle of same increase and difference decrease, and finally combine the above definition domain and monotonicity to obtain the value range.

The domain of the composite function is determined by both the inner and outer functions.

y=f(x), u=g(x) is known.

then f(g(x)) is called a composite function composed of f(x) and g(x), where f(x) is the outer function and g(x) is the inner function.

If we know that the domain of f(x) is (a,b), and we find the domain of f(g(x)), we only need to make a if we know that the domain of f(g(x)) is (p, q), and find the domain of f(x).

It is summarized by p: functions such as f(x), f(g(x)), f(h(x)) and other functions or composite functions, as long as the previous corresponding rule f is the same, then the definition domain is calculated as: the value range of the expression in parentheses after the corresponding rule f is the same, and the range of x can be obtained, which is the definition domain.

The domain of the composite function is determined by both the inner and outer functions.

For example, if y=f(x) and u=g(x) are known, then f(g(x)) is called a composite function composed of f(x) and g(x), where f(x) is the outer function and g(x) is the inner function.

If you know that the domain of f(x) is (a,b), and you need to make a <> to find the domain of f(g(x)).

First, find the definition domain of the outer function.

2. The value range of the inner function is determined according to the definition domain of the outer function, and the value range of the inner function in the composite function is the intersection of the definition domain of the outer function and the value range of the inner function itself.

3. According to the value range of the inner function, determine the inner function, that is, the definition domain of the entire composite function. For example, the function f(x)=ln(-x +9) can be seen as a composite of two functions, y=lnt and t=-x +9. So find the domain of this function as .

First, the definition domain of the outer function y=LNT is t 0 Second, the value range of the inner function t=-x +9 itself is t 9, so in this composite function, the value range of the inner function is the intersection of t 0 and t 9 0 t 9 Third, according to the value range of the inner function, the definition domain of the inner function is -3 x 3, and this is the definition domain of the entire composite function.

For example: f(x)=lg(x 2-1), the outer function is y=lgt, and the inner function is t=x 2-1>>

Looking at the outer function, when t>0, lgt r, so the range of this composite function is r

Here's a very simple example of a pickpocket, from which we can learn:

1.Use variables to decompose the composite function into several simple functions.

2.Define the domain and not forget.

3.First, find the range of the inner function.

4.Use the image to pick up the edge.

5.After finding the range of the outer function.

For the examination of the six basic elementary functions that students learned in high school, most of the test questions appear in the form of composite functions. There are three main types of related questions: function analytic problems, definition domain problems, value range problems, and monotonicity problems.

Among them, the definition domain and value range of composite functions are one of the hot issues in the high school mathematics examination, and it is also a place where students are prone to make mistakes. To solve such problems, we can start from the structural characteristics of the composite function and find out the key to determine the domain and value range of the composite function. Composite functions often appear in the form of (fg(x)), where g(x) is called the inner function and (fx) is called the outer function, for example, in the function y=a2x-(1a>0, and a≠1), the inner function is g(x)=2x-1 and the outer function is (fx)=ax.

1. Definition domain of composite function Example 1: Knowing the function (fx) = aunt%x, g(x) = log2x, find the domain of the composite function (fg(x)). According to the analysis of the law of composite function correspondence, the set of objects acted on by the inner function is its definition domain, and the object of (fg(x)) is the set composed of the same imitation x, so the definition domain of (fg(x)) and g(x) is the same set. SolutionBecause (fx)=aunt%x, so x Qinshan 0, and g(x)=log2x 0=log21, so the definition domain of (fg(x)) is [1, + according to the definition domain of the inner function is the definition domain of the compound friend middle function.

The domain of the composite function is determined by both the inner and outer functions.

Composite functions, where f (x) is the outer function and g (x) is the inner function. There are several important definitions to understand: defining domains, value ranges, composite functions.

Question Type 1:

Knowing the domain of the function y=f(x)[m,n], how to find the domain of the composite function y=f(g(x)))?

1. Idea analysis: This question type is the range of the independent variable x of y=f(x), and the range of the independent variable x of y=f(g(x)) is found, and the key is that g(x) of the composite function is equivalent to x of the function.

2. Solution strategy:

Let t=g(x), since the domain of y=f(x) is [m,n], so the domain of y=f(t) is also [m,n], i.e., t=g(x) m,n], so find the solution set of the inequality m g(x) n, that is, the domain of y=f(g(x)).

Example 1: Knowing the domain of the function y=f(x)[0,3], find the domain of the function y=f(3+2x).

Question Type 2:

Knowing that the compound abstract function y=f(g(x)) defines the domain [m,n], how to find the domain of the function y=f(x))?

1. Idea analysis: This question type is the range of the independent variable x of y=f(g(x)), and the range of the independent variable x of y=f(x) is found, and the key is that g(x) of the former is equivalent to x of the latter.

2. Solution strategy: let t=g(x), define the domain [m,n] according to the composite abstract function y=f(g(x)), and substitute the range of x into t= g(x), so as to find the range of the value of t, that is, the definition domain of y=f(x).

Example 2: Knowing the domain of the function y=f(2x-1) [0,3], find the domain of the function y=f(x).

Question Type 3 (a combination of the first two):

Given that the composite abstract function y=f(g(x)) defines the domain [m,n], how to find the domain defined by the composite abstract function y=f(h(x)))?

1. Idea analysis: This question type is the range of the independent variable x of y=f(g(x)), and the range of the independent variable x of y=f(h(x)) is found, the key is that g(x) of the former is equivalent to h(x) of the latter, so first find the definition domain of the "bridge" function y=f(x).

2. Solution strategy: Use the method of question type 2 to find the definition domain of y=f(x) according to the definition domain of y=f(x)), and use the method of question type 1 to find the definition domain of y=f(h(x)) according to the definition domain of y=f(x)).

Example 3: Knowing the domain of the function y=f(2x-1) [0,3], find the domain of the function y=f(3+x).

The domain of the composite function is determined by both the inner and outer functions.

For example, if y=f(x) and u=g(x) are known, then f(g(x)) is called a composite function composed of f(x) and g(x), where f(x) is the outer function and g(x) is the inner function.

If you know that the domain of f(x) is (a,b), and you need to make a <> to find the domain of f(g(x)).