1. Definition:
A power function is a function of the form f (x) = ax b, where a is constant and not equal to 0 and b is a real number. Exponential function refers to a function in the form of f (x) = a x, where a is a constant greater than 0 and not equal to 1, and x can be any real number.
2. Basic nature:
The basic properties of power function include:
When b is a positive real number, the power function is increasing function; When b is a negative real number, the power function is a decreasing function; When b is 0, the power function is always a constant function (unless a=0). The value of the power function at x=0 is 1. The domain of power function is all real numbers that make radix x b meaningful.
The basic properties of exponential function include:
The exponential function is increasing function. The exponential function takes the value of 1 when x=0. The domains of exponential functions are all real numbers.
3. Image features:
The image characteristics of power function and exponential function are different:
The image characteristics of power function depend on the positive and negative properties of power exponent b with x as the base. When b> 1, the power function rises rapidly on the right side of the X axis, and the curve gradually bulges upward; When 0
The image features of exponential function are gradually increasing, but the growth rate is decreasing. When the cardinality is greater than 1, the rising speed of the exponential function on the right side of the X axis decreases gradually. When 0
4. Function properties:
There are some different functional properties between power function and exponential function:
The power exponent in the analytic formula of power function can be any real number, so it can be used to represent the operation of non-integer power, such as square root, cube root and so on. But the power exponent of exponential function can only be a real number and cannot represent a non-integer power. The image of power function can contain the negative semi-axis of X axis, while the domain of exponential function only contains positive real numbers.