What is the monotonicity of composite functions?

Methods for judging the monotonicity of composite functions: 1. Derivatives 2. Constructing basic elementary functions (functions with known monotonicity)

3. Composite functions

According to the formula of same increase and different decrease, first judge the monotonicity of the inner function, and then judge the monotonicity of the outer function. In the same domain, if the two functions have the same monotonicity, then the composite function is an increasing function in this domain. Otherwise, it is a decreasing function.

4. Definition method

5. Combination of numbers and shapes

The monotonicity of a composite function generally depends on the monotonicity of the two functions contained in the function

(1) If both are increasing, then the function is an increasing function

(2) If one is decreasing and the other is increasing, then it is a decreasing function

(3 ) Both are subtractive, which is an increasing function

The derivative formula of a composite function

(3) F'(g(x)) = [ F(g(x) + dg(x)) - F(g(x)) ] /dx = [ F(g(x) + dg(x)) - F(g(x)) ] / dg(x) * dg(x)/ dx = F'(g) * g'(x)

The elective textbook for senior high school students includes derivatives and their applications, and a good grasp of the definition of monotonicity of functions.

The definition method is generally used to prove the monotonicity of a function.

For the inverse proportional function y=k/x, when x is greater than 0, Y decreases as x increases