In recent years, learning exercises have appeared in college entrance examination and scientific research papers, such as the following questions: f(x+T)=Tf(x), f(-x)=af(x)+b, which are actually extensions of functional properties. The source of the test questions is the periodicity of the function and the parity of the function, which is often the final question, and the proposition generally starts with a special one, such as f (x+t). Is x+T=Tx true for x∈R? Let's take x=0 to deny it. When dealing with such problems, candidates should contact the nature of the function they have learned, or start with images.

6. Mathematical thought:

(1) When solving function problems, it is sometimes necessary to combine the image of the function (not the proof); When discussing classification, we should determine the classification standard and follow the principle of no duplication and no leakage.

(2) The idea of function equation can solve the following problems: ① The equation with solution can be transformed into the domain for finding the function value through variable separation; ② The number of roots of the equation can be reduced to the number of intersections of images of two functions.

(3) The problem of constant establishment: by separating or transforming variables into unary functions, it is transformed into the problem of finding the maximum value of functions.

(4) converting multiplication operation into addition operation and converting complex operation into simple operation by taking logarithm; When drawing, you should know the growth rate of exponential function, power function and linear function (such as the number of solutions of equation x2=2x).