Liz S.

asked • 04/17/17

Express the​ function, f, in simplified form. Assume that x can be any real number.

f(x)=√100(x+3)^2

Misty S.

Is the square root sign over the whole equation or just over the 100?
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04/17/17

Liz S.

over the the whole equation
 
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04/17/17

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Kenneth S. answered • 04/17/17

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People seem to be saying equation when they should be saying EXPRESSION.
 
It's important to use grouping symbols to show precisely what is the scope of the radicand. I'll use square brackets:
 
Assuming that f(x)=√[100(x+3)^2] is intended, the simplification is f(x) = 10•|x+3|.
 
The absolute value bars are required.
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Arturo O.

Kenneth, why are the absolute value bars required when [-(x+3)]2 = (x+3)2 ?  Is that just to keep it single-valued?  I recall a physics problem in electrostatics (many years ago) where if you did not consider the negative square root, you lost part of the solution.  In fact, the professor mentioned in class that the most common mistake (which I made too) was not considering the negative square root (good old CSUN physics days!).
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04/18/17

Kenneth S.

It's a function with Real inputs and Real outputs, and by definition the square root of a quantity must be non-negative.
If there were a suitable domain restriction, the absolute value bars could be omitted. 
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04/18/17

Arturo O.

But my point is that the (x+3) factor is squared in the expression for f(x), and therefore the argument of the square root is positive, regardless of the sign of (x+3). You end up with a real value for f(x) regardless of the sign of (x+3).
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04/18/17

Kenneth S.

The request was to SIMPLIFY.  √[(x+3)2] is |x+3| because if we do not guarantee non-negativity, then the situation is that a negative result might have occurred (when x< -3) as a result of taking square root.
 
In Exercise sets in Algebra Ii/College Algebra textbooks, there is often an instruction to assume that all variables are non-negative, so that this absolute value nuance can be avoided.
 
It's not enough to just say that the square rooting cancels out the squaring! 
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04/18/17

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