- #1

CGandC

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- 24

- Homework Statement:
- Find if the following functions are injective and if so find their inverses.

- Relevant Equations:
- Familiarity with Lambda notation ( from lambda calculus )

I found the following functions ( In lambda notation ) to be injective, and now I am trying to find the inverse functions for them ( the inverse for the Image of ## f ## ) but I am stuck and I need help:

1. ## f = \lambda n \in \mathbb{N}. (-1)^n + n^2 ##

2. ## f = \lambda g \in \mathbb{R} \rightarrow\mathbb{R}. \lambda x \in \mathbb{R} . g(x+1) ##

1. ## f = \lambda n \in \mathbb{N}. (-1)^n + n^2 ##

2. ## f = \lambda g \in \mathbb{R} \rightarrow\mathbb{R}. \lambda x \in \mathbb{R} . g(x+1) ##

**In 1.**This function is indeed injective. My first approach to find the inverse was handwavingly done as follows ## (-1)^{2g(n)+1} + (2g(n)+1)^2 =n \longrightarrow g(n) = \frac{\sqrt{n+1} -1}{2} ## ( ## n \in \mathbb{N} ## ) , I've done this in order for ## (-1)^{2g(n)+1} ## to be equal to ## -1 ##, but clearly ## 2g(n) + 1 = \sqrt{n+1} ## and it won't always be the case that ## \sqrt{n+1} ## is a natural number so clearly, ## g ## is not the correct inverse function for ## f ##.**In 2.**I found this function to be injective ( maybe I am wrong? ) but then I was completely stumbled and don't know how to seek the inverse.