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?

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.

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

To solve the function g of f, you correctly substituted f(x) = x^2 + 1 into g(x), giving g(f(x)) = √(x^2 + 1). However, the simplification you made was inaccurate.

When you wrote √(x^2 + 1) as x + √1, you mistakenly assumed that (√a)^2 is equal to a for any positive number a. However, this assumption is not true in general.

To simplify √(x^2 + 1), you can apply a different property of square roots known as the square root property, which states that the square root of the square of a number is equal to the absolute value of that number. In this case, √(x^2) is equal to |x|.

Therefore, correctly simplifying √(x^2 + 1) yields |x| + 1. This explains why your book arrived at √(x^2 + 1) instead of x + √1.