Solution:
y=k/x
Assumption: x 1 < x2
Yes: y (x2)-y (x1) = k (1/x2-1/x1).
Namely: y (x2)-y (x1) = k (x1-x2)/[(x1) (x2)]
1, when k > 0, there are: y (x2)-y (x 1) < 0.
At this point, y is the subtraction function;
2. when k < 0, y (x2)-y (x 1) > 0.
At this point, y is a increasing function;
Method 2:
Solution:
y=k/x
y'=-k/x？
Obviously:
1, when k > 0 and y' < 0, y is a monotonically decreasing function;
2. When k < 0, y' > 0, y is a monotonically increasing function.
1 is a very simple question in senior one mathematics. A∩B means to find the intersection of A and B, that is, to find the same elements of A and B, which can be directly seen as 8 (please contact me if you have any questions, please adopt it if you are satisfied ~).
Senior one math (very simple) 0
y=ax^2+2x+ 1
The minimum value of is less than or equal to 0, and the opening of the image is upward.
So delta = 4-4a > = 0a <; = 1，a & lt0
So 0
1/a & gt； 1
(2π)/a & gt； 2π
High school math problem 1, very simple oblique two-way direct vision, high school solid geometry commonly used map.
The specific operation method is as follows
Based on the original graphic parameters
Keep the bottom edge of the graph unchanged.
Up to the original 1/2.
The 90-degree angle will automatically change to 45-degree angle.
In this way, you get an oblique two-metric direct view.
Four years older.
The abscissa of the vertex of a simple mathematical function in Senior One is: x =-(-2a)/2 = a.
When the vertex is on the left side of x=3, the problem condition is satisfied.
So: a≤3
Ask a simple math problem in senior one. From the meaning of the question, we can know that A = {x | x >-2, x ∈ r} B = {x | x < 1, x ∈ r}
Then A∪B=R, A∪B = {x |-2 < x < 1, x ∈ r}
Mathematics in senior one is a simple problem: A (- 1,-1) B (1, 3) C (x, 5) three-point * * line, so vector AB= vector BC (2 2,4) = (x-1.
Senior one math problem ... the properties of simple T-T inverse function.
(1) Two mutually inverse images are symmetrical about the straight line y = x;
(2) The necessary and sufficient condition for a function to have an inverse function is that the domain and value domain of the function are a mapping;
(3) The function and its inverse function are monotonously consistent in the corresponding interval;
(4) A general even function must have no inverse function (but a special even function has an inverse function, such as f(x)=a(x=0), and its inverse function is f(x)=0(x=a), which is a very special function). odd function does not necessarily have an inverse function. If a odd function has an inverse function, its inverse function is also odd function.
(5) All implicit functions have inverse functions;
(6) The monotonicity of continuous functions is consistent in the corresponding interval;
(7) The function of strict increase (decrease) must have the existence theorem of the inverse function of strict increase (decrease).
(8) The inverse function is mutual.
(9) Definition domain and value domain are opposites, and the corresponding laws are reciprocal (three opposites).
(10) Once the original function is determined, the inverse function is determined (three definitions).
The general form of exponential function is y = a x(a >；; 0 and not = 1), from the above discussion of power function, we can know that if X can take the whole set of real numbers as the domain, then we only need to make.
As shown in the figure, different sizes of a will affect the function diagram.
Exponential function:
In the function y = a x, you can see:
The domain of (1) exponential function is the set of all real numbers, provided that a is greater than 0 and not equal to 1. For the case that a is not greater than 0, there is no continuous interval in the definition domain of the function, so we will not consider it.
At the same time, a equal to 0 is generally not considered.
(2) The range of exponential function is a set of real numbers greater than 0.
(3) The function graph is concave.
(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 (of course, it can't be 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 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, then the function passes (0, 1+b).
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, the two functions are symmetric about y, but neither function has parity.
Conversion of cardinality:
For any meaningful exponential function:
Add a number to the index 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 move down.
Images of basis functions and exponential functions;
(1) From the point where the exponential function y = a x intersects the straight line x= 1 (1), on the right side of the y axis, the corresponding base of 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 bottom is big.
logarithmic function
Generally speaking, if the power of a (a is greater than 0, and a is not equal to 1) is equal to n, then this number b is called the logarithm of n with the base of a, and it is recorded as log aN=b, where a is called the base of logarithm and n is called a real number.
If the real number formula has no root number, then as long as the real number formula is greater than zero, if there is a root number, the real number is required to be greater than zero, and the formula in the root number is greater than zero.
Cardinality is greater than 0 instead of 1.
Why is the base of logarithmic function greater than 0 instead of 1?
In the ordinary logarithmic formula, a
The general form of logarithmic function is y=log(a)x, which is actually the inverse function of exponential function and can be expressed as x = a y, so the stipulation of a in exponential function is also applicable to logarithmic function.
The figure on the right shows the function diagram of different size A:
You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.
The domain of (1) logarithmic function is a set of real numbers greater than 0.
(2) The range of logarithmic function is the set of all real numbers.
(3) The function image always passes through the (1, 0) point.
(4) When a is greater than 1, it is a monotonically increasing function and convex; When a is less than 1 and greater than 0, the function is monotonous and concave.
(5) Obviously, the logarithmic function is unbounded.
Operational properties of logarithmic function;
If a > 0 and a is not equal to 1, m >；; 0, N>0, then:
( 1)log(a)(MN)= log(a)(M)+log(a)(N)；
(2)log(a)(M/N)= log(a)(M)-log(a)(N)；
(3) log (a) (m n) = nlog (a) (m) (n belongs to r)
Relationship between logarithm and exponent
When a is greater than 0 and a is not equal to 1, the x power of a =N is equivalent to log (a) n.
Mathematics in senior one is very simple ~ ~ f (-x) = 2 (-x- 1) 2-3.
=2(x^2+2x+ 1)-3
=2x^2+2x+2-3
=2x^2+2x- 1
f(x)=2(x^2-2x+ 1)-3
=2x^2-2x- 1
So it's odd and even.