April 20th, 2019, 08:32 PM  #11  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
What is $\sqrt{(4)^2}$? It's $4$. It is true that if $f(x) = x^2$ then $f^{1}(16) = \{4, 4\}$. In this case $f$ does not have a functional inverse so the notation $f^{1}$refers to the set inverse. This is the notation and the way you would think about it. [I hope my terminology is clear. The functional inverse of $f$ is a function $g$ such that $fg = gf = id$ where the concatenation of functions refers to function composition. $fg$ means first $g$ and then $f$; and $id$ is the identity function on fhe reals. And the set inverse of a number is the set of values that are mapped to it by $f$]. If $x$ is a real variable the symbol '$\sqrt{}$' is defined to be the positive of the two values given by the set inverse. That's what the notation means. It does not mean anything else. It does not mean the set inverse. It doesn't mean "really one thing but we say it's another." The ONLY thing it means is the positive of the two values given by the set inverse. If the variable $z$ is a complex number, we do not typically use the notation $\sqrt z$ because there is no notion of positive and negative numbers in the complex plane. We can of course choose a branch of a multivalued complex function. But we never use the square root sign. $\sqrt{x^2} = x$ and there is simply no other way to think about it that doesn't lead to confusion. If you want the set inverse, then write $f^{1}(16) = \{4, 4\}$. Last edited by Maschke; April 20th, 2019 at 09:10 PM.  
April 21st, 2019, 12:03 AM  #12 
Senior Member Joined: Oct 2009 Posts: 850 Thanks: 327  
April 21st, 2019, 10:10 AM  #13 
Newbie Joined: Apr 2019 From: Jor Posts: 5 Thanks: 0 
I'm not really into this advanced branch of math but for me, when he said √x² = x that seemed fine for me, maybe I didn't get what he really meant but whatever, I'm here for my basic question. in all the types of rational powers , when is it needed to use absolute values? or is sqrt(x^2) the only case?


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absolute, radicals, radicsls 
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