Inverse of a function property

In summary, the conversation discusses three questions related to the inverse of a function. The first two questions ask if $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$, and the third question asks if $f(f^{-1}(B))$ is true for a subset $B\subset R$. The speaker provides an example using the function $f(x)=x^2$ to show that the answer to the first two questions depends on the specific function and definition of $f^{-1}$. The speaker is unsure about their proof for the third question and asks for clarification. The conversation ends with a request for an explanation of the previous answer.
  • #1
OhMyMarkov
83
0
Hello everyone!

I have three questions:

(1) If $x\in R$, is it true that $f ^{-1} (f(x)) = x$?
(2) If $y\in R$, is it true that $f (f^{-1}(y)) = y$?
(3) If $B\subset R$, is it true that $f(f ^{-1} (B)$?

I think I have showed it for (3), but not sure of my proof. For (1) and (2), I considered the function $f (x) = x^2$. $f^{-1}(1)$ can be 1 and -1...

Thanks for the help!
 
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  • #2
The answer may depend on what \(f\) is and on the precise definition of \(f^{-1}\). Also, in (3), \(f(f^{-1}(B))\) cannot be true or false.
 
  • #3
OhMyMarkov said:
Hello everyone!

I have three questions:

(1) If $x\in R$, is it true that $f ^{-1} (f(x)) = x$?
(2) If $y\in R$, is it true that $f (f^{-1}(y)) = y$?
(3) If $B\subset R$, is it true that $f(f ^{-1} (B)$?

I think I have showed it for (3), but not sure of my proof. For (1) and (2), I considered the function $f (x) = x^2$. $f^{-1}(1)$ can be 1 and -1...

Thanks for the help!

As in Your example, the (1) supplies one and only one x if and only if $\displaystyle f^{-1} (*)$ is a single value function...

Kind regards

$\chi$ $\sigma$
 
  • #4
I've returned to although I have seen this before, and I thought I was convinced.

Could you explain your answer, I don't think I understand...

Thank you.
 
  • #5


Hello! I would like to clarify some points regarding the inverse of a function.

Firstly, the inverse of a function is defined as a function that "undoes" the original function. In other words, it takes the output of the original function and gives back the input. So, for (1) and (2), the statements are true if and only if $f$ is a one-to-one function, meaning that each input has a unique output.

Secondly, the statement in (3) is not always true. It depends on the specific function $f$ and the set $B$. If $f$ is not one-to-one, then $f^{-1}(B)$ may not be well-defined, and therefore, $f(f^{-1}(B))$ may not equal $B$.

For the example given, $f(x)=x^2$ is not a one-to-one function, so the statements in (1) and (2) are not always true. However, for (3), if we take $B=\{1\}$, then $f^{-1}(B)=\{1,-1\}$ and $f(f^{-1}(B))=\{1\}$. Therefore, it is not true that $f(f^{-1}(B))=B$ for all sets $B$.

I hope this helps clarify the concept of the inverse of a function. It is important to consider the properties of the specific function in order to determine the truth of these statements.
 

Related to Inverse of a function property

What is the inverse of a function?

The inverse of a function is a mathematical operation that reverses the output of a function back to its original input. In other words, if a function takes an input of x and produces an output of y, the inverse function will take an input of y and produce an output of x.

How do you find the inverse of a function?

To find the inverse of a function, you can follow these steps:1. Write the function in the form of y = f(x).2. Switch the x and y variables.3. Solve for y.4. Replace y with f^-1(x) to represent the inverse function.

What is the property of a function and its inverse?

The property of a function and its inverse is that when the two functions are composed together, they cancel each other out and return the original input. In other words, (f^-1 o f)(x) = x and (f o f^-1)(x) = x.

What is the notation used for inverse functions?

The notation used for inverse functions is f^-1(x). This is read as "f inverse of x" and represents the inverse of the function f.

Why is the inverse of a function important?

The inverse of a function is important because it allows us to solve equations with unknown variables. It also helps us to better understand the relationship between inputs and outputs in a function. Additionally, the concept of inverse functions is used in many areas of mathematics, such as calculus and geometry.

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