Is it true that only strictly monotonic functions have inverse functions?

If y = 1 / x, it is not monotone function, but there is inverse function. Some discontinuous functions also have inverse function

Is it true that only monotone functions have inverse functions

No, a sufficient condition for a monotone function to be an inverse function, y = 1 / x, is not a monotone function in the range of all nonzero real numbers, but there is an inverse function. You will know the necessary and sufficient condition for an invertible function to be an inverse function

If a function has an inverse function, is it necessarily monotone

Yes, because the definition of a function is that an independent variable corresponds to a function value (that is, an X corresponds to a Y)
The inverse function of a function is also a function, and it must satisfy the definition of function
So if it is not a monotone function (one y corresponds to multiple x), then an independent variable corresponding to multiple function values will appear after the inverse function (one X corresponds to multiple y)
Does not conform to function definition
So if a function has an inverse function, it must be monotonic

Why does monotone function have inverse function

It is not monotone function that has inverse function, as long as there is one-to-one correspondence between X and Y. for example, single point function or other functions constructed (discontinuous function is easy to construct), just like the representation of Wayne diagram in textbook
It should be said that monotone functions must have inverse functions, but functions with inverse functions are not necessarily monotonic, such as y = K / X

Finding the inverse function of y = x + (1 / x)

If the value range of X is not limited, the function has no inverse function
Let's help you find the inverse function of y = x + 1 / X (x > = 1). Other inverse functions of (0,1] or [- 1,0) or (- ∞, - 1] can be solved similarly
The denominator is removed to get YX = x ^ 2 + 1, and x ^ 2-y * x + 1 = 0,
According to the mean inequality, Y > = 2 * √ (x * 1 / x) = 2,
Therefore, the root formula of quadratic equation is used to obtain x = [y + √ (y ^ 2-4)] / 2 (Y > = 2),
By exchanging X and y, the inverse function of the function y = x + 1 / X (x > = 1) is y = [x + √ (x ^ 2-4)] / 2 (x > = 2)
Similarly, y = x + 1 / X (0

Inverse function of y = x / (x + 1) (x > - 1)

y=x/(x+1)
y(x+1)=x
xy+y=x
xy-x=-y
x(y-1)=-y
x=y/(1-y)
The inverse function is y = x / (1-x) x ≠ 1

The inverse function of (x) = 1

1. Solve X: x = (1-y) / Y from known function
2. Change the letter X to y and the letter Y to X: y = (1-x) / X
3. Get the definition domain: X ≠ 0
The inverse function of a given function is y = (1-x) / x, (x ≠ 0)

There is a mass of 0.2kg. The sliding friction coefficient between the block and the horizontal plane is 0.2. The sliding friction coefficient between the block and the horizontal plane is 0.2 (1) The distance that the block can slide on the horizontal plane; (2) The work done by the block to overcome the friction. (G is taken as 10m / S2)

(1) The kinetic energy theorem is applied to the whole process of block motion
0-0=mglsin30°-μmgs
The solution is: S = 10m
(2) In the whole process, the work done by friction force is w = - μ MGS
By substituting the data, w = - 4J
So the work done by the block to overcome the friction is 4J
Answer: (1) the distance that the block can slide on the horizontal plane is 10 m;
(2) 4 J. the work done by the block to overcome the friction

High school physics is really urgent online 1. The speed of manned spacecraft in near earth flight is about Km / S. (known radius of the earth r = 6400km, gravity velocity of the earth's surface g = 10m / S2). (v = 8.0km/s) 2. In order to make the spacecraft reach the above speed, an acceleration process is needed. During the acceleration process, the astronauts are in the The ratio of the gravity in this state to the gravity at rest on the earth's surface is called the tolerance value. If K is expressed, then k = (suppose the mass of the astronaut is m, and the acceleration of acceleration rise is a). When selecting an astronaut, the tolerance value of the astronaut in this state is required to be 4 ≤ K ≤ 12, indicating that the variation range of the acceleration value during the launch of the spacecraft is (overweight k = 1 + A / g 30m / S2 ≤ a ≤ 110m / S2) 4. When the spaceship enters the orbit of 3R from the earth's surface to make a circular motion around the earth, the astronauts with a mass of 64kg are in the State, his emphasis on N. The actual gravity is N. (total weightlessness? 0? 40)

1. The speed of manned spacecraft near the ground is the maximum surrounding velocity, that is, the first space velocity, which can be solved according to the centripetal force provided by local gravity
When mg = MV ^ 2 / R is substituted into the data, v = v = 8.0km/s can be obtained
2. In the process of acceleration, the support force is n-mg = ma, n = Mg + Ma, and the astronaut is overweight, which is the apparent weight
The gravity in this state is called the apparent weight, and the ratio of the gravity at rest on the earth's surface is called the tolerance value. If K is expressed, then k = n / Mg = (Mg + MA) / Mg = 1 + A / g
When selecting an astronaut, he is required to have a tolerance value of 4 ≤ K ≤ 12 in this state. When k = 4, a = 30m / S2, and when k = 12, a = 110m / S2. Therefore, the variation range of the acceleration value of the spacecraft during launch is 30m / s2 ≤ a ≤ 110m / S2)
4. When the spaceship enters the orbit of 3R from the earth's surface to make circular motion around the earth, the astronauts are already in circular motion. Gravity provides all the centripetal force, and is in complete weightlessness, and the apparent weight is zero
The actual gravity mg '(G' is the acceleration of gravity at 3R) at 3R above the ground is obtained from the joint solution of GMM / (4R ^ 2) = mg 'and GMM / R ^ 2 = mg
G '/ g = R ^ 2 / (4R) ^ 2, the apparent weight mg' = 64 * 10 / 16 = 40n was calculated as G '= g / 16

An object with mass m is thrown up vertically, and the air resistance is f in the process of moving in the air. The acceleration of the object in the process of ascending and descending is calculated A worker pushes a 45kg car horizontally with a horizontal thrust of 90N. At this time, the acceleration of the car is 1.8m. When the worker stops pushing the car after the car moves, what is the acceleration of the car,

1、 In the rising process, the motion direction is upward, the resistance is downward, and the gravity is downward. The acceleration is a = G + (F / M)
In the process of descent, the motion direction is downward, the resistance is upward, and the gravity is downward. The acceleration is a = g - (F / M)
2、 When a worker pushes a cart, the car is subject to thrust and friction
F-f=ma
That is, f = f-ma = 90n-45kg · 1.8m/s? 2 = 9N
When the car is not pushed, the car is only subject to friction, then the acceleration is:
a=f/m=0.2m/s²

Is it true that only strictly monotonic functions have inverse functions?