Exponential function image example The general form of the exponential function is y=a^x(agt; 0 and ≠1) (x∈R). It is one of the elementary functions. It is a monotonic, downwardly convex, and unbounded differentiable positive function defined in the real number domain.

Contents

Mathematical terms

...Translation of base:

Graph of base and exponential functions:

Comparison of powers:

Domain: set of real numbers

R range: (0, ∞)

Method of fractional simplification and skills

The correspondence between the image of the exponential function and the properties of the exponential function

Edit this paragraph of mathematical terminology

The exponential function is an important function in mathematics. This function applied to the value e is written as exp(x). It can also be written equivalently as e, where e is a mathematical constant, which is the base of the natural logarithm, approximately equal to 2.718281828, also known as Euler's number. The exponential function is very flat for negative values ??of x and climbs rapidly for positive values ??of x, equaling 1 when x equals 0. The slope of the tangent line at x is equal to the value of y there multiplied by lna. That is, from the derivative knowledge: d(a^x)/dx=a^x*ln(a). As a function of a real variable x, the graph of y=ex is always positive (above the x-axis) and increasing (looking from left to right). It never touches the x-axis, although it can get arbitrarily close to it (so, the x-axis is the horizontal asymptote of this graph. Its inverse function is the natural logarithm ln(x), which is defined for all positive x . Sometimes, especially in science, the term exponential function is used more generally for exponential functions of the form kax

where a is called the "base" and is any positive real number not equal to 1. This article initially focuses on the exponential function with the base Euler number e. The general form of the exponential function is y=a^x(agt; 0 and ≠1) (x∈R). From the above discussion on the power function. It can be known that in order for x to take the entire set of real numbers as its domain, it can only be seen in the figure that the different sizes of a affect the function graph. It can be seen in the function y=a^x: (1) Exponent. The domain of the function is the set of all real numbers. The premise here is that a is greater than 0 and not equal to 1. For the case where a is not greater than 0, there will inevitably be no continuous interval in the domain of the function, so we will not consider it. At the same time Functions with a equal to 0 are generally not considered. (2) The value range of the exponential function is the set of real numbers greater than 0. (3) Function graphs are all downwardly 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 rule, that is, when a goes from 0 to infinity, it is in the process of an exponential function (of course it cannot be equal to 0). , the curve of the function goes from the position of the monotonically decreasing function close to the positive half-axis of the Y-axis and the X-axis to the position of the monotonically increasing function close to the positive half-axis of the Y-axis and the negative half-axis of the X-axis respectively. The straight line y=1 is a transition position from decreasing to increasing. (6) The function always tends to the X-axis infinitely in a certain direction and never intersects. , (if y=a^x b, then the function passes through the point (0, 1 b) (8) Obviously the exponential function is unbounded. (9) The exponential function is neither an odd function nor an even function. (10) When two exponential functions When a in is the reciprocal of each other, the two functions are symmetric about the y-axis, but neither function has parity. (11) When the independent variables and dependent variables in the exponential function are mapped one by one, the exponential function has an inverse function. .

Edit this paragraph... Translation of the base:

For any meaningful exponential function: Add a number to the exponent, and the image will shift to the left; Subtract a number and the image will shift to the right. Adding a number to f(X) will shift the image upward; subtracting a number will shift the image downward. That is, "add up and subtract down, add left and subtract right"

Edit the image of the base and exponential function in this paragraph:

Exponential function

(1) From the exponential function y=a^x intersects the straight line x=1 at point (1, a). It can be seen that on the right side of the y-axis, the corresponding base of the image changes from small to large from bottom to top. (2) From the intersection of the exponential function y=a^x and the straight line x=-1 at the point (-1, 1/a), it can be seen that on the left side of the y-axis, the corresponding base of the image changes from large to small from bottom to top. (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 image is high"; on the left side of the y-axis, "the base is large and the image is low". (As shown on the right)》.

Edit this paragraph to compare the sizes of powers:

Common methods for comparing sizes: (1) Difference (quotient) method: (2) Function monotonicity method; (3) Intermediate value Method: To compare the sizes of A and B, first find an intermediate value C, then compare the sizes of A and C, B and C, and obtain the size between A and B based on the transitivity of the inequality. When comparing the size of two powers, in addition to the above general methods, you should also pay attention to: (1) For the comparison of the size of two powers with the same base and different exponents, you can use the monotonicity of the exponential function to judge. For example: y1=3^4, y2=3^5, because 3 is greater than 1, the function increases monotonically (that is, the greater the value of x, the greater the corresponding y value), because 5 is greater than 4, so y2 is greater than y1. (2 ) For the comparison of two powers with different bases and the same exponent, the exponential function

can be judged by using the changing rules of the exponential function image. For example: y1=1/2^4, y2=3^4, because 1/2 is less than 1, the function image monotonically decreases in the domain; 3 is greater than 1, so the function image monotonically increases in the domain, at x=0 The graphs of both functions pass through (0, 1). Then as x increases, the y1 graph decreases, while y2 increases. When x is equal to 4, y2 is greater than y1. (3) For different bases and different exponents To compare the sizes of powers, you can use intermediate values ??to compare. For example: lt; 1gt; For the size comparison of three (or more than three) numbers, you should first group them according to the size of the values ??(especially the size of 0 and 1), and then compare the sizes of each group of numbers. . lt; 2gt; When comparing the sizes of two powers, if you can make full use of "1" to build a "bridge" (that is, compare their sizes with "1"), you can quickly get the answer. So how to judge the size of a power and "1"? From the image and properties of the exponential function, it can be seen that "the same big but small".

That is, when the base a and 1 and the inequality sign between the exponent x and 0 are in the same direction (for example: a 〉1 and x 〉0, or 0 〈 a 〈 1 and x 〈 0), a^x is greater than 1, and when it is in opposite directions a^x is less than 1. 〈3〉Example: Are the following functions increasing or decreasing functions on R? Explain the reason. ⑴y=4^x Because 4gt;1, so y=4^x is an increasing function on R; ⑵y=(1/4)^x Because 0lt;1/4lt;1, so y=(1/ 4)^x is a decreasing function on R

Edit the domain of this paragraph: the set of real numbers

Refers to all real numbers

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

Edit the methods and techniques of fraction simplification in this paragraph

(1) Factor the numerator and denominator, and reduce the reducible ones first ( 2) Use the basic properties of the formula to transform complex fractions into simplified fractions, and transform different denominators into identical denominators (3) Simplify the appropriate fractions first and make key breakthroughs. Exponential functions

(4) You can consider the overall idea and use the substitution method to simplify the fraction

Edit the correspondence between the image of the exponential function and the properties of the exponential function in this paragraph

(1) Along the curve The definition domain of the 〈=〉 function extending infinitely to the left in the x-axis direction is (-∞, ∞). (2) The curve is above the x-axis, and infinitely depends on the exponential left or right as the x value decreases or increases. Function

Near the When 0, the function value y=a0 (zero power) = 1 (agt; 0 and a≠1) (4) When agt; 1, the curve gradually rises from left to right, that is, agt; when 1, the function is at (-∞ , ∞) is an increasing function; when 0lt; alt; 1 is, the curve gradually decreases, that is, when 0lt; alt; 1, the function is a decreasing function on (-∞, ∞).