How is the inverse solution found, and what is the method of solving the inverse function?

Generally, y=f(x) is converted into x=f(y), and then x and y can be swapped.

For example: y=ln(x) x=e y inverse function y=e xy=x x= y inverse function y= x

Generally speaking, if the range of the function y=f(x)(x a) is c, if we find a function g(y) where g(y) is equal to x, then the function x= g(y)(y c) is called the inverse of the function y=f(x)(x a), denoted as y=f-1(x).

The domain and range of the inverse function y=f -1(x) are the domain and domain of the function y=f(x), respectively. The most representative inverse functions are logarithmic and exponential.

To put it simply, the inverse function is to solve x from the function y=f(x), denoted by y: x= (y), if for every value of y, x has a unique value corresponding to it, then x= (y) is the inverse function of y=f(x), customarily, x is used to represent the independent variable, so x= (y) is usually written as y= (y) (i.e. swap x,y).

To find the inverse of a function:

1) Solve x from the original function formula and denote it by y;

2) Swap x, y, and 3) indicate the domain of the inverse function.

For example, find the inverse function of y= (1-x) Note: (1-x) means (1-x) under the root number

Solution: Both sides are squared, and y = 1-x

x=1-y²

Swap x, y gives y=1-x

So the inverse function is y=1-x (x 0).

Note: x in the inverse function is y in the original function, and y 0 in the original function, so x 0 in the inverse function

In the original function and the inverse function, since the positions of x and y are exchanged, the definition domain of the original function is the domain of the inverse function, and the value range of the original function is the definition domain of the inverse function.

There are two methods: 1: solve y=f(x), take x as the unknown, and solve x=g(y); Swap x,y,get:

The inverse function is y=g(x)2: swap x,y gets:x=f(y), with y as an unknown, and the solution y=g(x) is the inverse function.

In general, let the domain of the function y=f(x)(x a) be c, and according to the relationship between x and y in this function, x is represented by y, and x= g(y)If for any value of y in c, by x= g(y), x has a unique value in a corresponding to it, then, x= g(y) means that y is an independent variable, x is a function of the dependent variable y, and such a function y= g(x)(x c) is called the inverse function of the function y=f(x)(x a), which is denoted as y=f (-1) (x) The inverse function y=f (-1) (x) (x) defines the domain and the value range of the function y=f(x) respectively The sine function and its inverse function:

f(x)=sinx->f(x)=arcsinx cosine function and its inverse:f(x)=cosx->f(x)=arccosx tangent and its inverse:f(x)=tanx ->f(x)=arctanx cotangent and its inverse:

f(x)=cotx->f(x)=arccotxI hope it can help you, please give a "good review".

In general, let the domain of the function y=f(x)(x a) be c, and according to the relationship between x and y in this function, x is represented by y, and x= g(y)If for any value of y in c, through x= g(y), x has a unique value in a corresponding to it, then, x= g(y) means that y is an independent variable, x is a function of the dependent variable y, and such a function y= g(x)(x c) is called the inverse function of the function y=f(x)(x a), which is denoted as y=f (-1) (x) The domain of the inverse function y=f (-1) (x) (x) and the range of values are the domain and domain of the function y=f(x) respectively.

Example: Trigonometric functions.

The sine function and its inverse function: f(x)=sinx->f(x)=arcsinx cosine function and its inverse function: f(x)=cosx->f(x)=arccosx tangent function and its inverse function:

f(x)=tanx ->f(x)=arctanx

The cotangent function and its inverse function: f(x)=cotx->f(x)=arccotx, I hope it can help you, please give "good reviews".

Finding the inverse function.

From the previous example and the definition of the inverse function, it is not difficult to see that if you want to find the inverse function of the function y=f(x), you can follow the following steps:

determine the domain and value range of the function y=f(x);

Regard y=f(x) as the equation about x, and solve the equation to get x=f-1(y);

Swap x,y to obtain the analytic formula y=f-1(x) of the inverse function;

Write out the definition domain of the inverse function (the value range of the original function).

Here's how, please take the test:

If it helps, big and bold.

Please take the town.

f(x)=(2x +1)^3

2x +1)^3=y

2x+1= Occlusion y

2x=³√y-1

x=(³y-1)/2

The inverse function is f(x)=(in x-1) 2

f(x)=2√5x+8

2√5x+8=y

2√5x=y-8

x=(y-8)/2√5

5(y-8)/10

The inverse function is f(x) = macro roll 5(x-8) 10

y(x+1)=x-1 from the original formula, yx+y=x-1 by removing brackets, and yx-x=-1-y by moving the term, i.e., (y-1)x=-1-y, so x=1+y 1-y

Solution: (1) First, find the definition domain of the function. y=[1+ (1-x)] [1- (1-x)] 1-x 0, i.e. x 1;1- (1-x)≠0, i.e. x≠0; Define the domain x 1 and x≠0(2) to sort out the original function, so that y=[2-x+2 (1-x)] x let (1-x)=t, (t>0 and t≠1) then x=1-t 2, the original function is y=(1+t 2+2t) (1-t 2)=(1+t) (1-t) y(1-t)=1+t,y-yt=1+t,(y+1)t=y-1,t=(y-1) (y+1) 1-x)=(y-1) (y+1), x=1-[(y-1) (y+1)] 2 The inverse of the original function is y=1-[(x-1) (x+1)] 2,(y 1, and y≠0; x≠-1）

The domain of the inverse function of the function y=10 to the power of x is the domain of the original function y=10 x

1 value range (-1, positive infinity).

So the domain of the inverse function of y=10 to the power of x-1 is the range of values (-1, positive infinity).

How do I find the inverse of the function y=f(x)? Some books give the general steps:1determine the range b of the function y=f(x); 2.Solve from y=f(x).

x=g（y）；3.Exchange.

The position of x and y in x=g(y) gives y=g(x), and the function y=g(x) with b as the domain is the inverse function sought. We know that given a real number a, a is in the domain b of the function y=f(x) if and only if the corresponding equation a=f(x) has a solution in the domain of the function y=f(x).

Therefore, to determine whether A is in the range B of the function y=f(x) is to study the solvability of the equation A=F(X), and the judgment of this solvability, in many cases, depends on the solution of the equation A=F(X) itself, that is, whether the x in the equation A=F(X) can be really solved, that is, the root is found, which is equivalent to step 2 aboveConcretization or specialization.

So, we have reason to say step 1and step 2Sometimes it's simply an inseparable step.

So on what occasion step 1And the steps are really two steps? In cases where the range of the function y=f(x) is known, or when it can be easily determined by means of images, etc., these are two steps.