How do you calculate the problem of inverse functions? Any tips?

For example, read more books, look at the basics!!

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 x=f-1(y). The domain and domain of the inverse function x=f-1(y) are the domain and domain of the function y=f(x), respectively. Hall back.

Write y=f(x)=(x+13) (4x-1) and then denote x as y.

x+13=y*（4x-1）=4xy-y

4y-1）*x=y+13

x=（y+13）/（4y-1）

So, the inverse function y=(x+13) (4x-1) sine function and its inverse function: f(x)=sinx->f(x)=arcsinx cosine function and its inverse function: f(x)=cosx->f(x)=arccosx

The tangent function and its inverse function: f(x)=tanx->f(x)=arctanx cotangent modulo comma and its inverse function: f(x)=cotx->f(x)=arccotx

The exponential function and its inverse function: f(x)=a x->f(x)=logax

The way to find the inverse function is to swap x and y, and then solve y.

Introduction to Inverse Functions:

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

In general, if x corresponds to y with respect to some correspondence f(x), y=f(x), then the inverse function of y=f(x) is x=f-1(y). The condition for the existence of an inverse function (a single-valued function by default) is that the original function must be one-to-one (not necessarily within the entire number field). Note:

Superscript"−1"It refers to the power of the advertisement function, but not the exponential power.

Let the domain of the function y=f(x) be d and the range of values be f(d). If, for each y in the range f(d), there is only one x in d such that g(y)=x, then a function defined on f(d) is obtained according to this correspondence rule, and this function is called the inverse function of the function y=f(x).

From this definition, it can be quickly concluded that the definition domain d and the value range f(d) of the function f are exactly the value range and definition domain of the inverse function f-1, and the inverse function of f-1 is f, that is, the functions f and f-1 are inverse functions of each other.

In contrast to the inverse function y=f-1(x), the original function y=f(x) is called a direct function. The image of the inverse function and the direct function is symmetrical with respect to the straight line y=x. This is because, if (a,b) is any point on the image of y=f(x), i.e., b=f(a).

According to the definition of the inverse function, there is a = f-1(b), i.e. the point (b, a) on the image band of the inverse function y=f-1(x). The points (a,b) and (b,a) are symmetrical with respect to the straight line y=x, and from the arbitrariness of (a,b), we can know that f and f-1 are symmetrical with respect to y=x.

Write y=f(x)=(x+13) (4x-1);

Then denote x by y;

x+13=y*(4x-1)=4xy-y；

4y-1)*x=y+13；

x=(y+13)/(4y-1)

Then write x as f(x) (1) and y as x, and you get the inverse function.

So, the inverse function f (-1)=(x+13) (4x-1).

From the problem f(x)=(x+13) (4x-1) makes y=f(x).

Get y=(x+13) (4x-1).

This results in x=(y+13) (4y-1).

In this case, x=f(x) (1).

i.e. [f(x) (1)]=(x+13) (4x-1) specific steps:

Convert function to inverse function step:

1.Determine the range of the original function.

2.Solve the equation to solve x.

3.Swap x,y, specify the defined domain.

It is represented as follows in the coordinate diagram.

Write f(x) as y

Simplify: 4xy-y=x+13 (4y-1)x=13+y x=(13+y) (4y-1).

Replace x with f-1(x) and y with x as the inverse function.

Just swap x and y and write it in standard form.

To put it simply, the inverse function is to solve x from the function y=f(x) and use y to represent x= (y), if for each 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 position).

To find the inverse of a function:

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

2) Swap. x,y,(3) indicates the domain of the definition of the inverse function.

For example, find y= (1-x).

Note: (1-x) is the square of both sides under the root number (1-x), resulting in 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.

y=（e^x-e^_x)/2

2y=(e^2x-1)/e^x

e^2x-2ye^x-1=0

Let t=e x e 2x=t 2

t^2-2yt-1=0

The solution gives t=y+ (y 2+1) and since t=e x >0, t=y- (y 2+1) is rounded

i.e. t=e x=y+ (y 2+1).

x=ln（y+√(y^2+1) ）

So the inverse function of y=(e x-e x) 2 is y=ln(x+ (x 2+1)) x r

y>-1,y(1+ x)=1- x,y+1) x=1-y,x=(1-y) (1+y),x=[(1-y) (1+y)] 2,x,y is swapped to obtain the inverse function y=[(1-x) (1+x)] 2(x>-1).