Shanghai Experimental School East School Wang Yuerong

We call the allowable range of independent variables of a function the domain of this function. So how do you find the domain of a function?

1. When the analytical expression is algebraic, x is an arbitrary real number.

Example 1 Find the domain of the following function: (1)y=-5x2, (2) y=3x+5,

Solution: (1)x is a real number; (2)x is a real number.

2. When the analytical formula is a fraction, X is a real number with a denominator other than zero.

Example 2. Find the domain of the following function (1)y= (2) y=

Solution: (1)∵x- 1≠0 ∴ The domain of the function is a real number of x≠ 1

(2) The domain of the function ÷ 1+3x≠0∴ is a real number of x≠-. ,

3. When the analytical formula has even roots, x is a real number with a non-negative root number.

Example 3. Find the domain of the following function

( 1)y=，(2)y=，(3)y=

Solution: (1)∵3- x≥0, ∴x≤3.

(2)∫2x+4≥0 ∴x≥-2

⑶,∴x≥-4

4. When the analytical formula is a compound expression, first list the inequalities one by one, find out the allowable range of each part, and then find out their common parts.

Example 4. Find the domain of the following function

( 1)y= (2)y= (3)y= (4)y=

Solution: (1)∵ ∴ ∴, x≠4.

(2)∵ 1-5x & gt； 0∴x & lt； .

⑶∴x>； 2 and x≠3.

(4) ∵

5. When the analytical formula involves specific application problems, it depends on the specific application problems.

If a function is used to reflect a practical problem, the value of the independent variable must make the practical problem meaningful in addition to representing the digital sub of the function.

Example 5. Xiaoming bought pencils from 10 yuan, and the price of each pencil was 0.38 yuan. Xiao Ming * * * bought X, and the rest of the money is Y. Find the resolution function of Y about X and point out the range of X.

Solution: according to the meaning of the question, the resolution function of y about x is: y =10-0.38 x.

When y=0, that is, 10-0.38x=0, ∴ x=26.3.

Countability of pencils

The value range of ∴x is: 0.

Note: How to find the maximum value of x? When 10 yuan spends on pencils, that is, the remaining money is zero, X is the maximum. And considering the countability of pencils, x should be an integer.

Example 6. Given that the circumference of an isosceles triangle is 17cm, what is the functional relationship between its base length ycm and its waist length xcm? And point out the definition domain of the function.

Solution: from the meaning of the question: y+2x= 17.

∴y= 17-2x

∵y & gt； 0, namely17-2x > 0 ∴x<； 8.5

The sum of two sides of a triangle is greater than the third side.

∴x+x>； y，y= 17-2x。

∴2x>； 17-2x，x & gt4.25

The numerical range of ∴x is 4.25 cm.

Summary: When the cardinal number is Y and the waist length is X, the domain of X is perimeter.

Comments on several solutions of function value domain

Anhui liqingshe

Function is one of the important basic concepts in middle school mathematics, which is closely related to algebra, equations, inequalities, trigonometric functions, calculus and so on, and is widely used. Function has a strong foundation and many concepts, among which the domain, range and parity of function are one of the difficulties, and it is a common question in college entrance examination. The following is the solution of the function range, for example, as follows.

First, the observation method

By observing the definition domain and properties of the function, combined with the analytical formula of the function, the range of the function is obtained.

Example 1 Find the range of function y = 3+√ (2-3x).

Hugging: According to the nature of the arithmetic square root, first find the range of √ (2-3x).

Solution: From the nature of arithmetic square root, we know that √ (2-3x) ≥ 0,

Therefore, 3+√ (2-3x) ≥ 3.

The domain of the function is.

Comments: The arithmetic square root has double non-negativity, namely: (1) the non-negativity of the square root and (2) the non-negativity of the value.

This problem is solved by directly observing the properties of arithmetic square root. This method is simple and clear for finding the range of a class of functions, and it is an ingenious method.

Exercise: Find the range of the function y=[x](0≤x≤5). (Answer: the range is: {0, 1, 2, 3, 4, 5})

Two. Inverse function method

When the inverse function of a function exists, the domain of its inverse function is the range of the original function.

Example 2 Find the range of function y=(x+ 1)/(x+2).

Cuddle: first find the inverse function of the original function, and then find its domain.

Solution: Obviously, the inverse function of the function y=(x+ 1)/(x+2) is: x = (1-2y)/(y- 1), and its domain is a real number of y≠ 1.

Comments: Find the domain of the original function by inverse function method, provided that the original function has inverse function. This method embodies the idea of reverse thinking and is one of the important methods to solve mathematical problems.

Exercise: Find the range of the function y = (10x+10-x)/(10x-10-x). (Answer: The range of the function is {y ∣ y)

3. Matching method

When a given function is a quadratic function or a compound function that can be transformed into a quadratic function, the range of the function can be obtained by using the matching method.

Example 3: Find the range of function y = √ (-x2+x+2).

Pointing: Formulating the root number into a complete square number, and using the maximum value of quadratic function to find it.

Solution: From -x2+x+2 ≥ 0, we can know that the domain of the function is x ∈ [- 1, 2]. At this time-x2+x+2 =-(x-1/2) 2+9/4 ∈ [0,9/4].

∴ 0 ≤√-x2+x+2 ≤ 3/2, and the value range of the function is [0,3/2].

Comments: To find the range of a function, we should not only pay attention to the application of the corresponding relationship, but also pay special attention to the restrictive effect of the definition range on the range. Matching method is an important thinking method in mathematics.

Exercise: Find the range of function y = 2x-5+√ 15-4x. (Answer: the range is {y∣y≤3})

Four. Discrimination method

If it can be transformed into a fractional function or an irrational number function of a quadratic equation about a variable, the range of the function can be found by discriminant method.

Example 4 Find the range of the function y = (2x2-2x+3)/(x2-x+ 1).

Guidance: transform the original function into a quadratic equation with independent variables, and determine the value range of the original function by using the discriminant of the root of the quadratic equation.

Solution: Change the above formula to (y-2) x2-(y-2) x+(y-3) = 0 (*).

When y≠2, δ = (y-2) 2-4 (y-2) x+(y-3) ≥ 0, and the solution is: 2 < x ≤ 10/3.

When y=2, equation (*) has no solution. The range of the function is 2 < y ≤ 10/3.

Comments: Turn the functional relationship into a quadratic equation F(x, y)=0. Because the equation has a real number solution, its discriminant is non-negative, and the range of the function can be found. Often adapt to the functions of y=(ax2+bx+c)/(dx2+ex+f) and Y = AX+B √ (CX2+DX+E).

Exercise: Find the range of the function y = 1/(2x2-3x+ 1). (answer: the range is y ≤-8 or y>0).

Verb (abbreviation of verb) maximum method

For the continuous function y=f(x) on the closed interval [a, b], we can find the extreme value of y=f(x) on the interval [a, b] and compare it with the boundary value f(a). F(b), the maximum value of the function can be found, and the range of the function y can be found.

Example 5: (2x2-x-3)/(3x2+x+ 1)≤0, and x+y= 1 is satisfied. Find the range of function z=xy+3x.

Guidance: Find the range of the independent variable X according to the known conditions, eliminate the objective function and formula, and find the range of the function.

Solution: ∫3 x2+x+ 1 > 0. The above fractional inequality and inequality 2x2-x-3≤0 have the same solution, the solution is-1 ≤ x ≤ 3/2, and x+y= 1, y= 1.

∴z=-(x-2)2+4 and x∈[- 1, 3/2], the function z is continuous in the interval [- 1, 3/2], and we only need to compare the size of the boundary.

When x=- 1, z =-5; When x=3/2, z= 15/4.

The range of the function z is {z ∣-5 ≤ z ≤ 15/4}.

Comments: This question is to transform the range problem of the function into the maximum problem of the function. If the interval has a maximum value, the range of the function can also be obtained by finding the maximum value.

Exercise: If √x is a real number, the range of the function y=x2+3x-5 is ().

A.(-∞，+∞) B.[-7，+∞] C.[0，+∞) D.[-5，+∞)

(answer: d).

Mirror image method of intransitive verbs

By observing the image of the function, the range of the function is obtained by combining numbers and shapes.

Example 6 Find the range of the function y=∣x+ 1∣+√(x-2)2.

Pointing: according to the meaning of absolute value, remove symbols and convert them into piecewise functions to visualize them.

Solution: The original function is -2x+ 1 (x ≤ 1).

y = 3(- 1 & lt； x≤2)

2x- 1(x & gt； 2)

Its image is shown in the picture.

Obviously, the function value y≥3, so the function value range [3, +∞].

Comments: piecewise function should pay attention to the endpoint of the function. Images using functions

Finding the range of a function embodies the idea of combining numbers with shapes. Is an important way to solve the problem.

There are many methods to find the function value domain, and they are also suitable for finding the function value domain through inequality method, function monotonicity, method of substitution and other methods.

Seven. Monotone method

The evaluation domain is evaluated by monotonically increasing or decreasing functions in a given interval.

Example 1 Find the range of function y = 4x-√ 1-3x (x ≤ 1/3).

Hugging: the known function is a compound function, that is, g (x) =-√ 1-3x, y = f (x)+g (x), and its domain is x≤ 1/3. In this interval, the increase and decrease of the function are discussed respectively, so as to determine the value range of the function.

Solution: Let f (x) = 4x, g (x) =-√ 1-3x, (x ≤ 1/3), and we can easily know that they are increasing function in the definition domain, so y = f (x)+g (x) = 4x-√/kloc.

When the definition domain x≤ 1/3, Y≤ f (1/3)+G (1/3) = 4/3, it is also a increasing function, so the function domain is {y | y ≤ 4/3}.

Comments: Using monotonicity to find the range of a function is to find the given interval of the function or the implied interval of the function. Combined with the increase or decrease of the function, the value of the function at the end of the interval can be found, and then the value range of the function can be determined.

Exercise: Find the range of the function y = 3+√ 4-x (answer: {y | y ≥ 3})

Eight. Alternative method

Replace some quantities in the function formula with new variables, so that the function can be transformed into a function form with new variables as independent variables, and then find out the value range.

Example 2 Find the range of the function y=x-3+√2x+ 1

Hugging: the original function is transformed into a quadratic function of variables through substitution, and the value range of the original function is determined by the maximum value of the quadratic function.

Solution: Let t=√2x+ 1 (t≥0), then

x= 1/2(t2- 1).

So y =1/2 (T2-1)-3+t =1/2 (t+1) 2-4 ≥1/2-4 =-7/2.

So the range of the original function is {y | y ≥-7/2}.

Comments: Convert irrational function or quadratic function into quadratic function, and determine the range of the original function by finding the maximum value of quadratic function. This method of solving problems embodies the thinking method of method of substitution and induction. It is widely used.

Exercise: Find the range of the function y =√x- 1-x (answer: {y | y ≤-3/4}.

Nine. Constructive method

According to the structural characteristics of the function, the geometric figure and the combination of number and shape are given.

Example 3 Find the range of function y=√x2+4x+5+√x2-4x+8.

Pointing: deform the original function to construct a plane figure, and determine the range of the function with geometric knowledge.

Solution: The transformation of the original function is f (x) = √ (x+2) 2+1+√ (2-x) 2+22.

Make a rectangular ABCD with a length of 4 and a width of 3 and cut it into 12 units.

Square. Let HK=x, then ek=2-x, KF=2+x, AK=√(2-x)2+22.

KC=√(x+2)2+ 1 .

According to the triangle trilateral relationship, AK+KC≥AC=5. When a, k and c are three points * * *

Draw a line with an equal sign.

The domain of the original function is {y | y ≥ 5}.

Comments: For the function y = √ x2+a √ (c-x) 2+b (a, b and c are all positive numbers), it can be intuitive, clear, convenient and simple by constructing geometric figures. This is the combination of numbers and shapes.

Exercise: Find the range of function y=√x2+9 +√(5-x)2+4. (Answer: {y | y ≥ 5 √ 2})

Proportional method

For solving the range of a kind of conditional function, the condition can be transformed into a proportional expression and substituted into the objective function, and then the range of the original function can be obtained.

Example 4: Given x, y∈R, 3x-4y-5=0, find the range of function z=x2+y2.

Hugging: Transform the conditional equation 3x-4y-5=0 into a proportional formula, set the parameters and substitute them into the original function.

Solution: It is transformed from 3x-4y-5=0 and (x3)/4 = (y- 1)/3 = k (where k is the parameter).

∴x=3+4k,y= 1+3k，

∴z=x2+y2=(3+4k)2+( 14+3k)2=(5k+3)2+ 1。

When k =-3/5, x = 3/5 and y =-4/5, zmin= 1.

The value range of the function is {z | z ≥ 1}.

Comments: This question is a multifunctional relationship, which generally contains constraints. By setting parameters, the original function can be transformed into a single function. This problem-solving method embodies many thinking methods and has a certain sense of innovation.

Exercise: given x, y∈R and satisfying 4x-y=0, find the range of function f (x, y) = 2x2-y (answer: {f (x, y) | f (x, y) ≥ 1}).

XI。 polynomial division

Example 5 Find the range of the function y=(3x+2)/(x+ 1).

Hug: Transform the original fractional function into the sum of an algebraic expression and a fraction by long division.

Solution: y = (3x+2)/(x+1) = 3-1/(x+1).

∫ 1/(x+ 1)≠0, so y≠3.

∴ All real numbers whose range of function y is y≠3.

Note: This method can be used for functions in the form of y=(ax+b)/(cx+d).

Exercise: Find the range of the function y = (x2-1)/(x-1) (x ≠1). (Answer: y≠2)

Twelve. Inequality method

Example 6 Find the range of function Y=3x/(3x+ 1).

Hugging: First, find the inverse function of the original function, and construct inequalities according to the range of independent variables.

Solution: It is easy to find that the inverse function of the original function is y=log3[x/( 1-x)].

According to the definition of logarithmic function, x/( 1-x) > 0.

1-x≠0

0 < x

The range of the function (0, 1).

Comments: Investigate the value range of function independent variables, construct inequalities (groups) or important inequalities, find the function domain, and then find the domain. Inequality method is an important tool to solve problems and is widely used. It is one of the methods to solve mathematical problems.

The following is for practice: Find the range of the following functions.

1.y = √( 15-4x)+2x-5； ({y|y≤3})

2. y = 2x/(2x-1). (y>1or y