I had to solve the function g of f where f is x^2+1 and g is sq. root of x. I wrote this out as the sq. root of x^2+1. The ^2 and the sq. root canceled each other out, so I was left with x + sq. root of 1. But my book got sq. root of x^2+1. Was the further reducing I did acceptable or inaccurate?

Question

I had to solve the function g of f where f is x^2+1 and g is sq. root of x.  I wrote this out as the sq. root of x^2+1.  The ^2 and the sq. root canceled each other out, so I was left with x + sq. root of 1.  But my book got sq. root of x^2+1.  Was the further reducing I did acceptable or inaccurate?

jim · Answer

You can't separate the added parts and take the square root of each. (You could if they were multiplied, but not if added.)

I will show a counterexample.

You were looking at sqrt(x^2+1).

Consider this example - let x=3, then we have :

sqrt(3^2+1) = sqrt(9+1) = sqrt(10)

Now, you would have said that this was equal to

3 + sqrt(1) = 4.

But 4 is not the square root of 10.

We can get in deeper about why your approach doesn't work if you like, but this counterexample is enough to show it's not true.

Adriana · Answer

Thank you :D That really cleared it up! I never had anyone explain that concept so well before.