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 [3, +∞].

Comments: The arithmetic square root has double nonnegativity, that is, the nonnegativity of (1) root.

(2) The value is non-negative. This problem can be 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})

Second, the 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 = (10 ∧ x+10 ∧-x)/(10 ∧-x). (Answer: the range of the function is {y ∣ y 1})