Simplifying polynomial fractions - Equality question

I am studying the simplification of polynomial fractions. My textbook has the example of:

$\displaystyle \dfrac{2x+6}{x^2-9} = \dfrac{2}{x-3}$ : factored out (x+3) from the numerator and denominator

My confusion is over the sense of 'equality' of the 2 expressions. I expected to be able to use either expression interchangeably as the definition for a function. However,

$\displaystyle \ f(x) = \dfrac{2x+6}{x^2-9}$ is undefined for x = { 3, -3 }

$\displaystyle \ f(x) = \dfrac{2}{x-3}$ is undefined for x = { 3 }

The domains of the functions differ. It appears that 'information' was lost in the simplification of the original expression.

Are the 2 expressions not really "equal"? Am I misunderstanding something about the concept of equality with regard to expressions?

Re: Simplifying polynomial fractions - Equality question

Quote:

Originally Posted by

**ForumUser2** Are the 2 expressions not really "equal"?

That's right. As numbers, $\displaystyle \dfrac{2x+6}{x^2-9}$ and $\displaystyle \dfrac{2}{x-3}$ are equal iff $\displaystyle x\ne-3$. To compare these expressions as functions, we must equip each of them with a domain. As long as those domains do not include -3, they are equal as functions (in the set-theoretic sense). With their natural, i.e., maximal, domains, they are different as functions because their domains are different.

In the process of transformations (e.g., when solving an equation), it is important to check if each equality is a true identity, i.e., holds for all values of variables. Similarly, it is important to check if two equalities or inequalities are truly equivalent, i.e., are both true or both false for all values of variables.

Re: Simplifying polynomial fractions - Equality question

I just love your blog.Thanks for posting it.i have something that someone comeback again….there is a lot of useful information a person can get from here…Web DevelopmentI must say,well done...i visited it daily,what a colourful blog it is....colours attracted me a lot...Free Legal Advice.i visited it daily,what a colourful blog it is.... a very decent and nice blog...i like it so very much...