Generally speaking, matching method can be used to solve the problem of evaluation domain of quadratic function.

Example 1, the range of solution

Solution:

So the range of values is.

Second, the inverse function method

Generally speaking, the reciprocal relationship between the domain and the value domain of the original function and the inverse function can be used.

Example 2. Find the range of function.

Solution: Yes, because, so.

So the scope of this function is

Third, the separation constant method

Generally speaking, for fractional functions, a constant can be separated to find the range of the function.

Example 3, the value range of the solution

Solution:

but

That's right, so

In other words, the scope of the function is.

Note: Example 2 can also use the separation constant method to evaluate the domain. Interested readers can try it.

Four. Discrimination method

Generally, it is transformed into a quadratic equation about y, and the equation has a real number solution to find y.

Example 4, the range of the solution.

Solution: Remove the denominator.

that is

When y=2, this equation has no real root.

When this equation is a quadratic equation with one variable,

that is

So, because, then,

Therefore, the scope of this function is

Note: The following two points cannot be directly judged.

1, domain minus infinite points. 2. The numerator and denominator contain common factors.

Verb (abbreviation of verb) substitution method

The general shape is that by substitution (pay attention to the range of t at this time)

A numerical range was found in Example 5.

Solution: Order

So =

When t=0, the minimum value of y is 3.

So the range of values is.

Six, classified discussion method

Evaluate the domain by discussing the symbol of function definition domain X by classification.

A numerical range was found in Example 6.

Solutions;

Because, therefore, that is

while

that is

To sum up: the scope of.