What Functions Satisfy (f(x))^2 = x^2 and Are Continuous?

[Kate2010]

Homework Statement

Find 5 different functions f: R -> R such that (f(x))² = x²

How many continuous functions satisfy the requirement? Justify your answer.

Homework Equations

The Attempt at a Solution

So far I have:
f(x) = x
f(x) = -x
f(x) = |x|

Could I also have, for example, f(x) = (x² - 5x)/(x-5) as this cancels down to f(x)= x but is undefined at 5?

And I'm not sure how to answer the continuity part, so far all of the functions I have found are continuous (I think?). However, not all continuous functions satisfy it.

 

[Mark44]

What does your notation (f(x))₂ = x₂ mean?

 

[Kate2010]

Oh I'm sorry I typed it incorrectly. I meant squared but did subscript not superscript, it's edited now.

 

[Mark44]

I think that the fourth function you list satisfies the requirement that (f(x))² = x², so it should be easy to get one more.

The first three functions you listed are continuous, but the fourth one isn't, because it isn't continuous at x = 5.