The form of exponential function is y = a x, and 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.
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 ex, where e is a mathematical constant and the base of natural logarithm, which is approximately equal to 2. 7 1828 1828, also known as Euler number.
The image of exponential function is monotonous, always in the first and second quadrants, passing through the point (0, 1); Power function needs specific analysis of specific problems.
Exponential function: the position of the independent variable x in the exponent, y = a x(a >；; 0, a is not equal to 1), when a >; At 1, the function is increasing function, y >；; 0; When 0; 0.
Power function: the position of independent variable x in radix, Y = X A (A is not equal to 1). A is not equal to 1, but it can be positive or negative. Different values make different images and properties.
2. Different in nature
Power function attribute:
(1) positive attribute
When α >; 0, the power function y=xα has the following properties:
A, the images all pass through the point (1,1) (0,0);
B, the function in the image is the increasing function in the interval [0, +∞);
C, in the first quadrant, α >; 1, the derivative value increases gradually; When α= 1, the derivative is constant; 0 & ltα& lt； 1, the derivative value gradually decreases and approaches 0;
(2) negativity
When α
A, the images all pass through the point (1,1);
B, the image is a decreasing function in the interval (0, +∞); (Content supplement: If it is X-2, it is easy to get that it is an even function. Using symmetry, the symmetry axis is the Y axis, and the image can be monotonically increased in the interval (-∞, 0). The same is true for other even functions).
C, in the first quadrant, there are two asymptotes (coordinate axes), the independent variable approaches 0, the function value approaches +∞, the independent variable approaches +∞, and the function value approaches 0.
(3) Zero value attribute
When α=0, the power function y=xa has the following properties:
The image with y=x0 is a straight line with y= 1. Delete a point (0, 1).
Its image is not a straight line.
Exponential function properties:
The domain of (1) exponential function is r, where 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 it will not be considered, while 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)a & gt； 1, the exponential function increases monotonically; If it is 0
(5) It can be seen that when a tends to infinity from 0 (not equal to 0), the function curves tend to approach the positions of monotonic decreasing function and monotonic increasing function of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. Positive semi-axis of Y axis and negative semi-axis of X axis. The horizontal 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 exponential function is unbounded.
(8) Whether the exponential function is odd or even.
Exponential function has inverse function, which is logarithmic function and multi-valued function.
Monotone interval of 2 power function
When α is an integer, the positive and negative parity of α determines the monotonicity of the function:
① When α is a positive odd number, the image monotonically increases in the domain of R;
② When α is a positive even number, the image monotonically decreases in the second quadrant and monotonically increases in the first quadrant;
③ When α is a negative odd number, the image monotonically decreases in each of the first three quadrants (but it cannot be said that it monotonically decreases in the definition domain R);
④ When α is a negative even number, the image increases monotonously in the second quadrant and decreases monotonously in the first quadrant.
When α is a fraction (and the numerator is 1), the parity of α' s sign and denominator determines the monotonicity of the function:
(1) when α >; 0, when the denominator is even, the function monotonically increases in the first quadrant;
② When α >; 0, when the denominator is odd, the function monotonically increases in each of the first three quadrants;
③ When α
④ When α