Mirror method: use the rise and fall of function mirror to judge the monotonicity of function. Compound function method: using the property of "same increase but different decrease" to judge the monotonicity of compound function f[g(x)].
Specifically, the definition method is to judge the monotonicity of the function by taking the value, calculating the difference, deforming, determining the number and drawing the conclusion; The function property method can judge the monotonicity of complex functions by using the monotonicity of simple functions; The mirror method is to judge the monotonicity of the function according to the rise and fall of the mirror image of the function; The method of compound function is to judge the monotonicity of compound function according to the property of "same increase but different decrease"
It is also a common method to judge the monotonicity of relatively complex functions by using the monotonicity of common simple functions. For example, the monotonicity of quadratic function F (x) = AX 2+BX+C can be judged according to the sign of A, the position of symmetry axis and the direction of opening.
Mirror method is a method to judge the monotonicity of function by using the rise and fall of function mirror. If the function monotonically increases in a certain interval, then its image should rise in that interval; If a function monotonically decreases in a certain interval, then its image should decrease in that interval.
For example, for the function y=sinx, its image is wavy, that is to say, in each period, the value of the function increases first and then decreases, then increases and then decreases ... so we can say that the function is not monotonous in the whole definition domain.
Compound function method
The method of compound function is to judge the monotonicity of compound function according to the property of "same increase but different decrease" If the composite function composed of two functions is monotonically increasing, then the monotonicity of the two functions should be the same; If the monotonicity of the composite function decreases, then the monotonicity of the two functions should be opposite.
For example, for the compound function y=logax n and function y=logax and function y = x n, if a >: 1 and n>0, the compound function is monotonically increasing; If it is 0
To sum up, judging the monotonicity of a function needs to be comprehensively analyzed by combining definition, properties, images and composite functions. Only by mastering various methods can we solve related problems better.