My Math Forum Radicals and absolute value

 Pre-Calculus Pre-Calculus Math Forum

April 20th, 2019, 08:32 PM   #11
Senior Member

Joined: Aug 2012

Posts: 2,357
Thanks: 740

Quote:
 Originally Posted by Pasta Right, thanks, and same applies to the other situations i mentioned right? (even power with even root if the result is an odd power) No he's right @Maschke
No, he's wrong.

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 multi-valued 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

Quote:
 Originally Posted by Pasta No he's right @Maschke
He's very wrong.

 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?

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post corleoneee Calculus 5 October 22nd, 2013 12:31 AM shiseonji Algebra 2 September 24th, 2013 08:36 AM ohaider Algebra 6 February 6th, 2012 07:13 PM jonquil Algebra 6 April 13th, 2011 03:40 PM Mr_Quick Algebra 3 May 14th, 2009 07:50 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top