Do Factoring and Simplification Affect the Domain of a Function?

[Ryuzaki]

Suppose I have a function f defined on x, f(x) = (x^2)/x and another function g defined on x, g(x) = x. Are both these functions the same?

I mean, when you try to determine the Domain of a function, do you simplify it as much as possible, and then find the Domain? Or find the Domain on the face of the function?

In this case, what I think is that f has Domain-->ℝ-{0}, while g has Domain--> ℝ. Is this correct?

 

[micromass]

No, they are not the same. Indeed, f is not defined in 0, while g is.

However, f(x) and g(x) are the same for all nonzero x.

 

[dextercioby]

But of course, you can simplify without altering domains.

$$f(x) = g(x), ~ \forall x, ~ \mbox{if} ~ f(x)=1 ~ \mbox{and} ~ g(x) = \frac{x^2 +1}{x^2 +1}$$

Bottom line, find the domain and see if it changes by any possible factoring/simplification.