Ex is an exponential function. Exponential function is one of the important basic elementary functions. Generally y=ax (a is a constant, take a >；; 0, a≠ 1) is called exponential function, and the definition domain of the function is r. In the definition expression of exponential function, the coefficient before ax must be the number 1, and the independent variable x must be in the position of exponent, and it cannot be any other expression of x, otherwise it is not an exponential function. The function applied to the value e is written as exp(x). It can also be written as ex, 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.
Exponential function definition:
1, the domain of exponential function is r, where the premise is that a is greater than 0 and not equal to 1. For the case that A is not greater than 0, it will inevitably make the definition domain of the function discontinuous, so we will not consider it, and the function with A equal to 0 is meaningless and generally will not be considered.
2. The range of exponential function is (0, +∞).
3. The function graph is concave.
4. When a> 1, the exponential function increases monotonically; If it is 0
5. We can see an obvious law, that is, when a approaches infinity from 0 (not equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and X axis respectively, and monotonic increasing functions 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.
Introduction to ex:
Its image is monotonically increasing, x∈R, y >；; 0, which intersects the Y axis at (0, 1), the image is above the X axis, and the second quadrant is infinitely close to the X axis. Solution: y=ex is an exponential function of natural logarithm e and exponent x, and e is about 2.87 >; 1 monotonically increasing.
Ex parity:
Ex is neither a odd function nor an even function. F(x)= ex, f(-x)= e-x, -F(x)=- ex, f(x)≠f(-x)≠-f(x) Therefore, f (x) is a nonsingular non-even function.
Odd function introduced:
Odd function means that any x in the definition domain of the function f(x) whose definition domain is symmetrical about the origin has f(-x)=-f(x), so the function f(x) is called odd function. In odd function f(x), the signs of f(x) and f(-x) are opposite and their absolute values are equal, that is, f (-x) =-f (x); On the other hand, the function f(x) satisfying f(-x)=-f(x) must be odd function.
Characteristics of odd-numbered functions:
1, the odd function image is symmetric about the origin.
2. The domain of odd function must be symmetrical about the origin (0,0), otherwise it cannot be odd function.
3. If f(x) is odd function and is valid when x=0, then f(0)=0.
4. Let f(x) be differentiable in the domain. If f(x) is a odd function in the domain, then f 1(x) is an even function.
Introduction to even functions:
Generally speaking, if any x in the definition domain of function f(x) has f(x) = f (-x), then function f(x) is called EvenFunction.
Even function algorithm:?
1, the sum of two even functions is an even function.
2. The sum of two odd function is odd function.
3. The sum of an even function and a odd function is a non-odd function and non-even function.
4. The product obtained by multiplying two even functions is an even function.
5. The product obtained by multiplying two odd function is an even function.
6. The product of an even function multiplied by a odd function is odd function.
7. On the symmetric interval, the definite integral whose integrand is odd function is zero.
Function parity determination:
1, look at the image, odd function is symmetrical about the origin; Even functions are symmetric about y; That is, odd and even are functions about the origin and y axis symmetry, and only have constant functions, which are 0; Odd or even functions are functions that are neither symmetric about the origin nor symmetric about the Y axis.
2. See if certain conditions can be met. Odd function satisfies f (-x) =-f (x) for x in any domain; Even function satisfies f (-x) = f (x) for x in any domain; That is, even numbers and odd numbers, which satisfy f(-x)=f(x) and f(-x)=-f(x). For X in any field, it is just a function with a constant of 0. Non-odd and non-even, the pair f(-x)=f(x) and f(-x)=-f(x) does not hold in any domain.
Odd function even function algorithm;
1, the sum of two even functions is an even function.
2. The sum of two odd function is odd function.
3. The sum of an even function and a odd function is a non-odd function and non-even function.
4. The product obtained by multiplying two even functions is an even function.
5. The product obtained by multiplying two odd function is an even function.
6. The product of an even function multiplied by a odd function is odd function.
7. odd function must satisfy f(0)=0, because the expression f(0) means that 0 is within the defined range, and f(0) must be 0, so odd function does not necessarily have F(0), but when it has F(0), F(0) must be equal to 0, and it does not necessarily have f(0)=0, so the odd function is deduced. In this case, the function is not necessarily odd function.
8. The odd function f(x) defined on R must satisfy f (0) = 0; Because the domain is on r, there is f(0) at x=0. If you want to be symmetrical about the origin, you can only take a y value at the origin, which can only be f(0)=0. This is a conclusion that can be directly used: when x can be taken as 0 and f(x) is odd function, f(0)=0.