Prove domain and whether the function - f(x) = 1/(1 + |x|) is one to one. I am a first year Computer Science student studying Calculus and Linear Algebra this semester.
I have got two questions:

*

*how to find domain regarding finding the domain of the following function.

*Is this function one to one?

Question - 1 - Domain - My understanding
f(x) = 1/(1+ |x|)


*

*1 + |x|, denominator can never be 0.

*|x| is an absolute value, therefore it can be any number.

*Therefore the domain is (-α, α).

However, the solution is as follows and I do not understand this solution.
1 + |x| >= 1 

1/(1 + |x|) <= 1

0 < 1/(1 + |x|) <= 1 for all x E (-α, α)


Question - 2 - One to One function or Not
y = 1/(1 + |x|)

1 + |x| = 1/y

|x| = 1/y -1
x = +- (1/y -1)

From here, I can see that y must be greater than 0 since the denominator cannot be 0.


I get stuck here. I had to use wolframalpha site to plot the graph. Without using this website, how should I procced to test for one-to-one function?
Graph Plot to check one-to-one function
My understanding is that since there are two x values (+-) for each corresponding y, it is not one to one. Is my understanding correct?
Thanks very much.
https://www.wolframalpha.com/input/?i=f%28x%29+%3D+1%2F+%281+%2B+%7Cx%7C%29

The domain is $(-\infty, \infty)$ as you have observed. The function is not one-to-one because $f(-1)=f(1)$.
