Prove that function is invertible

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If this is the case, then you can use the Fundamental Theorem of Arithmetic to prove that the function is injective, and use some basic algebraic manipulations to show that it is also surjective. In summary, the function ${2}^{m}.(2n+1)-1$ from $\Bbb{N} \times \Bbb{N} \implies \Bbb{N}$ is invertible by applying the Fundamental Theorem of Arithmetic and using algebraic manipulations.
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Nath1
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prove that the function

${2}^{n}.(2n+1)-1 $ from $ \Bbb{N}$x$\Bbb{N}\implies\Bbb{N}$
is invertible

I know that a function to be invertible must be injective and surjective, I am not sure how to calculate this since in this case I need a pair (x,y) since the function comes from $ {\Bbb{N}}^{2}$.

Can anyone help me ? thanks
 
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  • #2
Nath said:
prove that the function

${2}^{n}.(2n+1)-1 $ from $ \Bbb{N}$x$\Bbb{N}\implies\Bbb{N}$
is invertible

I know that a function to be invertible must be injective and surjective, I am not sure how to calculate this since in this case I need a pair (x,y) since the function comes from $ {\Bbb{N}}^{2}$.

Can anyone help me ? thanks
Hi Nath, and welcome to MHB!

I think you need to double-check that you have read the question correctly. It only makes sense if there are two different variables in the function, perhaps ${2}^{\color{red}m}.(2n+1)-1 $.
 

FAQ: Prove that function is invertible

1. How do you prove that a function is invertible?

To prove that a function is invertible, you must show that it has a unique inverse function. This means that for every input in the original function, there is only one output, and vice versa. You can also use the horizontal line test to determine if a function is one-to-one, which is a necessary condition for invertibility.

2. What is the significance of a function being invertible?

An invertible function allows you to find the original input value when given the output value, making it useful for solving equations and finding unknown values. It also allows for the composition of functions, as the inverse function can "undo" the original function's actions.

3. Can all functions be inverted?

No, not all functions can be inverted. A function must be one-to-one, meaning each input has a unique output, in order to have an inverse. Some functions, such as exponential and logarithmic functions, are not one-to-one and therefore do not have an inverse.

4. What is the notation for an inverse function?

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

5. How can you check if a function is invertible algebraically?

To check if a function is invertible algebraically, you can use the process of finding the inverse function. This involves switching the variables x and y and solving for y. If you are able to isolate y and express it as a function of x, then the original function is invertible. Additionally, you can use the derivative of the function to determine if it is invertible, as a one-to-one function will have a non-zero derivative for all values in its domain.

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