Do Factoring and Simplification Affect the Domain of a Function?

  • Thread starter Ryuzaki
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In summary, the functions f(x) = (x^2)/x and g(x) = x are not the same, as f is not defined at 0 while g is. However, they are equivalent for all nonzero values of x. When determining the domain of a function, it is important to simplify it as much as possible while still considering any restrictions on the domain. In this case, the domain of f is ℝ-{0}, while the domain of g is ℝ. Simplifying the functions does not change the domain, but it is important to check for any possible changes when factoring or simplifying.
  • #1
Ryuzaki
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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?
 
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  • #2
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.
 
  • #3
But of course, you can simplify without altering domains.

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

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

1. Are (x^2)/x and x the same?

No, (x^2)/x and x are not the same. (x^2)/x is equivalent to x, but they are not identical expressions.

2. Can (x^2)/x and x be simplified to the same expression?

Yes, (x^2)/x can be simplified to x by canceling out the x in the denominator, but they are still not the same expression.

3. Is (x^2)/x always equal to x?

No, (x^2)/x is only equal to x when x is not equal to 0. Division by 0 is undefined, so the two expressions are not always equal.

4. Why is (x^2)/x not the same as x?

Even though (x^2)/x simplifies to x, they are not the same expression because x^2/x is a fraction while x is a single variable. They may represent the same value, but they are not identical.

5. Can (x^2)/x and x be used interchangeably in equations?

No, (x^2)/x and x cannot be used interchangeably in equations. They may have the same value, but they are not equivalent expressions and may result in different solutions when substituted into an equation.

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