Exponential function is an important function in mathematics. This function applied to the value e is written as exp(x). It can also be equivalently written as e, where e is a mathematical constant and the base of natural logarithm, which is about equal to 2.7 1828 1828, also known as Euler number. The exponential function is very flat for the negative value of x, and rises rapidly for the positive value of x, and is equal to 1 when x is equal to 0. The slope of the tangent at x is equal to y times lna. That is, from the derivative knowledge: d (a x)/dx = a x * ln (a). As a function of the real variable x, the image of y=ex is always positive (above the x axis) and increasing (from left to right). It never touches the X axis, although it can be anywhere near it (so, the X axis is the horizontal asymptote of this image. Its inverse function is the natural logarithm ln(x), which is defined on all positive numbers X. Sometimes, especially in science, the term exponential function is more generally used for any positive real number whose exponential function is 1 in the form of kax. Firstly, this paper mainly studies the exponential function based on Euler number e. The general form of exponential function is y = a x(a >；; 0 and ≠ 1) (x∈R). From the above discussion about power function, we can know that if X can take the whole set of real numbers as the domain, only the size of A as shown in the figure will affect the function diagram. In the function y = a x, we can see that the domain of (1) exponential function is the set of all real numbers, and the premise here is that a is greater than 0 and not equal to 1. For the case that a is not greater than 0, there will be no continuous interval in the definition domain of the function, so we will not consider it. The function with a equal to 0 is meaningless and generally will not be considered. (2) The range of exponential function is a set of real numbers greater than 0. (3) Function graphs are all convex. (4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.
5) We can see an obvious law, that is, when a tends to infinity from 0, it is an exponential function.
In this process (of course, it can't be equal to 0), the curve of the function tends to approach the position of the monotonic decreasing function of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing. (6) Functions always infinitely tend to a certain direction on the X axis and never intersect. (7) The function always passes (0, 1). (If y = a x+b, the function passes through the point (0, 1+b). (8) Obviously, the exponential function is unbounded. (9) Exponential function is neither odd function nor even function. (10) When a in two exponential functions is reciprocal to each other, the two functions are symmetric about y, but neither of them has parity. (1 1) When the independent variable and the dependent variable in the exponential function are mapped one by one, the exponential function has an inverse function.
Functions, where a is called "base", are not equal.