Used for simple analytic expressions.

Y = 1-√ x ≤ 1, range (-∞, 1)

Y = (1+x)/(1-x) = 2/(1-x)-1≦-1,range (-∞,-1.

2. Matching method

Mostly used for quadratic (type) functions.

Y = x2-4x+3 = (x-2) 2-1≥-1,range [- 1, +∞).

Y = e 2x-4e x-3 = (e x-2) 2-7 ≥-7, range [-7, +∞)

3. Alternative methods

Mostly used for compound functions.

Through substitution, higher-order function reduction, fractional function algebra and irrational function rationalization, it is convenient to go beyond the evaluation domain of function algebra.

Pay special attention to the range of intermediate variables (new quantities).

y=-x+2√( x- 1)+2

Let t=√(x- 1),

Then t ≤ 0 and x = t 2+ 1.

Y =-T2+2t+1 =-(t-1) 2+2 ≤1,range (-∞,1).

4. Inequality method

Using the basic properties of inequality is also a common method to find the domain.

y=(e^x+ 1)/(e^x- 1)，(0 & ltx & lt 1).

0 & ltx & lt 1,

1 & lt； e^x<； e，0 & lte^x- 1<； e- 1，

1/(e^x- 1)>； 1/(e- 1)，

y= 1+2/(e^x- 1)>； 1+2/(e- 1)。 Range (1+2/(E- 1), +∞).

5. Maximum value method

If the function f(x) has a maximum value m and a minimum value m, then the range is [m, M].

So the method of finding the domain is the same as the method of finding the maximum value.

6. Inverse function method

Some are also called inverse solutions.

The definition domain and value domain of a function and its inverse function are interchangeable.

If the range of a function is not easy to find, but the domain of its inverse function is easy to find, then we can get the former by finding the latter.

7. Monotonicity method

If f(x) is a increasing function in the domain [a, b], the range of values is [f(a), f(b)]. The range of subtraction function is.

[f(b)，f(a)]。 If you want a range, you must first find the domain. If it is a parabola, you must also see whether the vertex is within the definition domain.