I presented the following example in my Precalculus classes this past week and it bothered students:
Let
and
. Compute
and
and state the domain of each.
As usual, I’ll give you a second to think about it yourself.
..
..
Done yet?
.
Here are the answers:
The reason that the first one was unsettling, I think, is because of the restricted domain (despite the fact that the simplified form of the answer seems not to imply any restrictions).
The reason the second one was unsettling is because they had forgotten that 
Random Walks 
How did the class discussion go after you presented this problem? Did you address the square root issue? How did you have the class visualize the functions? Were these functions presented as part of a larger problem? Just some random questions that your post sparked. Thanks.
I am a fan of trying to visualize such functions, though I didn’t do it here. And they weren’t part of a larger problem. But both are good ideas! 🙂
Sorry…didn’t fully answer your questions:
The discussion went well, though it took some convincing. For the first function, I remind them that
is the function that was asked for, which clearly has domain no larger than 
. The fact that we simplify it is our business. The problem didn’t ask you to do that (though I do encourage it!).
For the second function, we did the problem carelessly first and I let them, intentionally, get the wrong answer
. Then I said…wait a second, 
only outputs positive numbers, but 
sometimes outputs negative numbers. Something went wrong. Then I remind them that actually, what they’re looking at is a situation where we have 
.
I like that you let them get the wrong answer, then you changed the format of the answer and asked them to check it for reasonableness. Getting students to check their answers is a challenge!
The whole sqrt(x^2) = |x| is very confusing for students. I’ve noticed that many algebra books don’t even try to deal with this: when they start talking about fractional exponents, they will just say the the base b is assumed to be non-negative. When I was in grad school for mathematics, I even heard a fellow grad student say, “My students don’t even know that the square root of x^2 is x,” and my office mate and I jumped all over this immediately.
I have a hypothesis that if you asked HS math teachers what the square root of x^2 is that probably 90% would say “x.” If you replied, “Are you SURE?” then most of them would realize their mistakes,
This is a symptom of a deeper problem: people want desperately to believe that all functions have inverse functions when they don’t. Instead, we make up what I like to call pseudo-inverse functions like sqrt(x) and sin^(-1)(x) that don’t behave the way inverse functions really do. I think if we could do a better job calling attention to this it would really help students.
A hearty “amen” to all you’ve said here, Harold!
It gets worse ! Defining sqrt{x^2} as |x| is dangerous for a good reason.
The absolute value of a signed number is an unsigned number, as it is a measurement. When you get to complex numbers the whole idea of positive and negative disappears, and the absolute value is more clearly a measurement. Applying the definition above to sqrt((1+i)^2) gives the result sqrt(2), which is unfortunate, to say the least. To convert the correct definition of sqrt(x^2) as the positive square root into “…which is |x| ” is careless, and overloads the symbolism, apparently so that we can have an inverse for f(x) = x^2