Radical in Function

May 2018
United States
So I have solved the following problem, but I don't understand the underpinning "why" of a particular part. I am speaking of part a. in the problem

We have function f composed of function g, which will look like this

after plugging g(x) into f(x).
I know (or at least I'm pretty sure) that my radical would simplify in the next step to \(\displaystyle |7-x|+6\) if I weren't dealing with a function, and I was told no other specifics about what could or could not be inside the radical. The index of the radical is even, the power it is raised to is even, and the result is odd.
The book tells me not to use absolute value however when removing the binomial from the radical, but rather just parentheses.
Now I know that I am dealing with a function's radical, and the domains of radicals exclude imaginary numbers produced by negative values inside of radicals, but I don't quite see the reasoning here. Assuming I am on the right course of reasoning, can someone connect the dots for me?
As far as I'm concerned, I'm not solving for the domain of the radical in that particular step, so I don't see why the way that I pull the value inside the radical to the outside should be affected by it. :(
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Forum Staff
May 2007
If $x\le 7$, then $\sqrt{7-x}$ is real and the square $7-x\ge 0$.
If $x\gt 7$, then $\sqrt{7-x}$ is imaginary and the square $7-x\lt 0$.
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