1. ## proof question

I'm assuming that the second sentence in each of these paragraphs is a proof, but I have no idea how to comprehend them. Help please.

2. Hello khor
Originally Posted by khor

I'm assuming that the second sentence in each of these paragraphs is a proof, but I have no idea how to comprehend them. Help please.
I'm not quite sure what you mean by "... is a proof." The second sentence in the first paragraph (and the third sentence in the second paragraph) is a statement, followed by an explanation of why that statement is true.

The first one is:
Statement

• The range of $y = x^2$ is $[0,\infty)$

Explanation

• The square of any real number is non-negative (in other words the range can't contain any negative numbers)

• Every non-negative number is the square of its own square root (in other words, every non-negative number is included in the range because there will always exist a corresponding element in the domain, namely, its square root)
The second one is:
Statement

• The range of $y = 1/x$ ... is the set of all non-zero real numbers.

Explanation

• $y = 1/(1/y)$ (in other words, the reciprocal function is its own inverse. So, since the domain of $y = 1/x$ is all non-zero real numbers the range will also be all non-zero real numbers)
I hope that is what you wanted.

(and the third sentence in the second paragraph)

Ooops, I mean third sentence there yeah.

(in other words, every non-negative number is included in the range because there will always exist a corresponding element in the domain, namely, its square root)

This is much clearer to me than what is in the book. What confused me was that the book phrases this in such an awkward way that even trying to understand this simple tautology from there gives me a headache. What the hell is that about? :/

(in other words, the reciprocal function is its own inverse. So, since the domain of $y = 1/x$ is all non-zero real numbers the range will also be all non-zero real numbers)
I'm having trouble with the concept of inverse right now but this does make a little sense.

Thanks for the explanations.