What is the inverse of h(y) where y=|x|

[matrix_204]

I'm preparing for my Statistics and Probability exam tomorrow, and I have a quick question:

What is the inverse of h(y) where y=|x|. (just to make sure, h'(x)=1, right?)

 

[0rthodontist]

h'(x) is not 1 for all x. Draw the graph of |x| and think again. Also do you know the horizontal line test for whether a function is invertible?

 

[matrix_204]

Well when I draw the graph for |x| i get like a graph like this starting at the origin \|/ , and I'm not sure how to find the inverse or by using the horizontal line test? is that like one-to-one function type?

 

[matrix_204]

This is the problem that I'm doing:
Suppose that Z is a standard normal random variable: i.e. Z~N(0,1).
a) Find the distribution of X=|Z| .
b) What is the density of X?
c) Find the distribution of Y=X^2
d) What is the joint distribution of X and Y?
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For a) P(X</=x) = P(|Z|</= x) = P(-z</= x </= z) = P(x</=z) - P(x</= -z)
I'm stuck here...

For b) is the density function for this the same as the one that is given as the definition. I mean fx(x)=[1/root2(pie)]e^(-x^2)/2?

For c) I got N(0, 2root(y)) as the distribution of Y=X^2.