1, domain and value domain: for power function f (x) = x n, where n is a real number, the domain is all real number x, and when n is a positive even number, the value domain is a positive real number set; When n is a negative even number, the range of values is positive real number and zero; When n is positive odd number or negative odd number, the range is real number.
2. Parity: When n is an even number, the power function is an even function symmetrical about y, that is, f (x) = f (-x); When n is odd, the power function is a odd function symmetric about the origin, that is, f(x)=-f(-x).
3. Monotonicity: According to the value of the exponent n of the power function, the monotonicity of the function can be determined. When n >; 0, the power function is increasing function; When n
4. limit: the limit of power function when x tends to positive infinity or negative infinity depends on the value of index n ... when n >; At 0, f(x) tends to positive infinity when x tends to positive infinity and to zero when x tends to negative infinity; When n
5. Asymptote: There may be a horizontal asymptote in the power function image (when n >; Y=0 is the horizontal asymptote when 0), and the vertical asymptote (when n
Many mathematical problems can be solved by using these power functions.
Practical application of power function
1, physics: power function is used to describe some physical phenomena, such as the relationship between force and acceleration in Newton's second law, the relationship between current and voltage in resistance, etc.
2. Biology: In biology, power function is widely used to describe the growth, metabolism and energy consumption of organisms. For example, the relationship between the volume and weight of an organism, and the relationship between energy consumption and weight can all be expressed by power functions.
3. Computer science: In algorithm analysis and performance evaluation, power function is often used to describe the relationship between the time complexity and space complexity of the algorithm and the input scale.