The general form of exponential function is y = a x(a >；; 0 and ≠ 1) (x∈R). This is a basic function. It is a monotone, convex, unbounded and differentiable positive function defined in real number field.

catalogue

Mathematical term

..... Basic translation:

Images of basis functions and exponential functions:

Power contrast:

Domain: real number set

R range: (0, +∞)

Methods and skills of fractional simplification

Correspondence between exponential function images and exponential function properties

Edit the mathematical terms in this paragraph.

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 natural logarithm ln(x), which is defined on all positive numbers X. Sometimes, especially in science, the term exponential function is more commonly used for exponential functions like kax.

Function, where a is called radix, is any positive real number that is not equal to 1. 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.

Edit this paragraph ... base translation:

For any meaningful exponential function: add a number to the exponent and the image will move to the left; Subtract a number and the image will move to the right. Add a number after f(X), and the image will shift upwards; Subtract a number and the image will pan down. That is, "addition, subtraction, multiplication and division, left plus right subtraction"

Edit this image of base and exponential function:

exponential function

(1) From the point where the exponential function y = a x intersects the straight line x= 1 (1), it can be known that on the right side of the y axis, the base corresponding to the image changes from bottom to top. (2) From the point (-1, 1/a) where the exponential function y = a x intersects with the straight line x=- 1, it can be seen that on the left side of the Y axis, the corresponding base of the image changes from large to small. (3) The relationship between the base of the exponential function and the image can be summarized as follows: on the right side of the Y-axis, the base is large and the graph is high; On the left side of the Y axis, "the bottom is big and the picture is low". (as shown on the right).

Edit the power comparison of this paragraph:

Common methods for comparing sizes: (1) ratio difference (quotient) method: (2) function monotonicity method; (3) Intermediate value method: compare the sizes of A and B, first find an intermediate value C, then compare the sizes of A and C and B, and get the size between A and B from the transitivity of inequality. When comparing the sizes of two powers, in addition to the above general methods, we should also pay attention to: (1) The comparison of the sizes of two powers with different indices at the same base can be judged by the monotonicity of the exponential function. For example: y 1 = 3 4, y2 = 3 5, because 3 is greater than 1, so the function monotonically increases (that is, the greater the value of x, the greater the corresponding value of y), because 5 is greater than 4 and y2 is greater than y 1. (2) For two powers with different cardinality and the same exponent, it can be exponential.

According to the changing law of exponential function image. For example: y 1 = 1/2 4, y2 = 3 4, because1/2 is less than1,the function image monotonously decreases in the definition domain; 3 is greater than 1, so the function image monotonically increases in the definition domain. When x=0, both function images pass (0, 1). Then with the increase of x, the image of y 1 drops, while y2 rises. When x is equal to 4, y2 is greater than y 1. (3) Different cardinality leads to different indices. For example, when the base number a and 1 are in the same direction as the inequality between the indices x and 0 (for example, a > 1 and x > 0, or 0 < a < 1 and x < 0), a x is greater than 1, and a x is less than 1. Explain why. (1) y = 4 x because 4 >; 1, so y = 4 x is the increasing function on R; (2) y = (1/4) x because 0

Edit the field of this paragraph: real number set

Refers to all real numbers

Edit the R range of this paragraph: (0, +∞)

Methods and skills of editing this simplified score

(1) Factorization of numerator and denominator, reducible first reduction (2) Using the basic properties of the formula, complex fractions are transformed into simplified fractions, and different denominators are transformed into the same denominator (3) Simplify some true fractions first, focusing on breaking through exponential functions.

(4) On the whole, method of substitution can be used to simplify the score.

Edit the correspondence between the exponential function image and the exponential function attribute in this paragraph.

The (1) curve extends infinitely to the left along the X axis. The domain of < = > function is (-∞,+∞). (2) The curve is above the X axis, and it is infinitely dependent on the exponential function with the decrease or increase of the X value to the left or right.

The range of the function < = > near the X axis (X axis is the asymptote of the curve) is (0, +∞) (3) When the curve passes through the fixed point (0, 1) < = > X = 0, the function value y=a0 (zeroth power) =1(a > 0 and a ≠1) (4) a > A > 1, the curve gradually rises from left to right, that is, A >: at 1, the function is increasing function at (-∞,+∞); 0<a< 1 means that the curve gradually decreases, that is, 0.