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## Operations of Functions! (Division) And some other stuff...

How would you solve this f(x)⁄g(x) problem (please include ALL steps and an explanation for each)?

f(x)=(5x+11)/(x2-9)
g(x)= (4x-2)/(x2-9)

Also, would the domain end up being (-∞, -3) U (-3, 1/2) U (1/2, 3) U (3, ∞)? Or would there be no 1/2? From what I think I got, the denominator should be 2(2x-1), so the denominator couldn't be 0, so it can't be 1/2. Am I right? Or would we just be looking in the middle of the solving/simplifying?

One last thing: when do we look in problems like these to get our domain (like when do we look to see what values are not included)?

Thanks!

"... would the domain end up being (-∞, -3) U (-3, 1/2) U (1/2, 3) U (3, ∞)?"

Yes, it would!!

If you have variable in denominator, we must indicate the domain of a giving expression, or equation, or function!!!

So, the first step will be to find the domain of f(x) and g(x),

x2 - 9 ≠ 0 ----> x ≠ ± 3.

Thus, both functions are defined for

x ∈ ( - ∞, - 3) U (- 3, 3) U (3, ∞)

........

After you finish division, you will insert another critical point x = 1/2 into domain of original functions.

However, in algebra, the priorities are...
SIMPLIFY
FACTOR
EVALUATE
SOLVE
Evaluation of domain does not come before simplification.
In any branch of mathematics the first priority is to identify the critical points in order to avoid any mistake.
Then the algebraic processes should be...
EVALUATE
SIMPLIFY
FACTOR
SOLVE
I found out that Nataliya was right; we find any values that do not fit the domain throughout, so the 1/2 would be correct in the domain. So I guess we'd have to go through the WHOLE process and catch any critical numbers, or numbers that don't fit the domain. Is that correct?
Thank you for the update.
It is correct, if, as I explain above, the algebraic processes are deemed incorrect.
Hi M;
I APOLOGIZE.  I FORGOT ABOUT THE SECOND HALF OF THE EQUATION.  HERE IT IS...
f(x)/g(x)=f(x) times 1/g(x)
(5x+11)    (x2-9)
(x2-9)       (4x-2)

I took g(x) and flipped the numerator and denominator to make this a multiplication rather than division question.

In this equation (x2-9) is in both the numerator and denominator.  These therefore cancel.

(5x+11)
(4x-2)

THE ONLY THING LIMITING THE CURRENT DOMAIN IS THAT X CANNOT = 1/2 TO RENDER THE NUMERATOR ZERO.

PLEASE ASK YOUR INSTRUCTOR IF WE SIMPLIFY AND THEN EVALUATE DOMAIN, OR IF WE EVALUATE DOMAIN BEFORE SIMPLIFICATION.  I HAVE RESEARCHED THIS AND CANNOT FIND AN ANSWER.