Basic algebra question

1/((x+1)/(x+2)) is equivalent to the reciprocal of (x+1)/(x+2), which is (x+2)/(x+1).

from the text, I have to replace all the X in F(x) with G(x) meaning

F(g(x)) is what I'm looking for and it turns out F(x) is 1/x and G(x) is (X+1)/(X+2)

so then it becomes 1/(x+1)/(x+2) but I don't know if and why I should put parenthesis to make it become like this 1/[(x+1)/(x+2)]

Basically I'd like to know the common rules in those type of situation with the fact that I know that dividing something means multiplying with the inverse of the divisor.

you realy should put some paranthesis...

what you thought about is:
(1/(x+1)) / (x+2) = 1 / ((x+1)*(x+2))

what the book meant is:
1 / ((x+1)/(x+2)) = (x+2) / (x+1)


why is this so? the philosophical abstract algebra behind it is deep but the for calculations you need only remember the method which is best explain by an example:
when you devide by 3 (=3/1), its the same as multipling by 1/3 (third)
same way when you devide by 2/3, its the same as multipling by 3/2

so when you do 1 devided by (x+1)/(x+2) its the same as multipling 1 by (x+2)/(x+1).


the same logic worked for your understanding of the problem
deviding (1/(x+1)) by (x+2) is the same as multpling by 1/(x+2)
so the result is (1*1) / ((x+1)*(x+2))

I hope I was clear...
if not ask again and ill try to rephrase

Iftah.

So if I follow what you said Iftah,
my reasoning is good for the technique but I got confused because I forgot to make proper use of parenthesis because the problem was 1/x and replacing that x byt (x+1)/(x+2) making it in reality something like

1/[(x+1)/(x+2)]

them meaning that this would be the same as (1/1)*[(x+2)/(x+1)] which in term would become

(x+2)/(x+1) < final answer

is that right ?

just read the posts that were posted while i wrote my first reply (hate when that happens)

what does G(F(X)) means? its a new function, different from F and from G.
the new function first activates F on X, then activates G on the result.
so you see if you want to write a formula for G(F(X)) you need to put F(X) in place of Y, but you must surround it with paranthesis to make sure it is calculated first.

*always* when you replace X in F(X) think of it as surrounded by paranthesis.
as you saw, without it the results are often a mistake

for example:
F(X) = X+1
G(Y) = 2*Y

G(F(X)) = 2*(X+1) = 2*X + 2

without paranthesis you would get
G(F(X)) = 2*X+1 <--- wrong!!!


Iftah.

[Edited by - Iftah on September 12, 2005 8:49:49 AM]

Ok I get it now

so the only problem I had was considering the parenthesis by forgetting them or ignoring them for that matter since I thought it didn't make a difference by I see it does

alright I understood

so something like X / Y / Z if there are no parenthesis means

X * 1/Y * 1/Z which would give X/Y*Z

but it would have been otherwise if there would have been parenthesis like in my problem

well, there is no meaning to A/B/C since you cant tell if its (A/B)/C or A/(B/C)

and these two are different:

(A/B)/C = A/(B*C)
A/(B/C) = (A*C)/B

without the paranthesis you guessed one option and did the calculations correct, but you guessed the wrong one.

I feel I wasnt clear on the explaination of G(F(X))...

anyways, if you are not sure if you got the right formula for G(F(X)) you can always check yourself -
choose X (say 2) then activate F on it (put 2 in F's formula) then activate G on the result (put the result in G's formula).
then try putting 2 in the formula you got for G(F(X)). it should come out the same.
(ofcourse getting the same doesnt prove you got it right...)


Iftah.

[Edited by - Iftah on September 11, 2005 10:00:25 PM]

The thing you got me to check from now on is the importance of parenthesis so I will make sure I make good use of them when they are there to avoid being confused by something like that again. by you explained properly it's just I'm tired and it's late and I should be going to bed instead of being hard-headed and willing to get to understand it.

When I don't understand something I can't sleep at night but now it's alright I understand where I made the mistake and by doing the problem again I see I'm getting the proper results.

Thanks !

always happy to help :)

im going to sleep also, good luck on your homework


Iftah.

I think you've already got it, but think of it this way just in case:

(a/b)/c = (a/b)*(1/c) = a/(bc)

However

a/(b/c) = a*(1/(b/c)) = a*(c/b) = ac/b

In your case, you have

1/((x+1)/(x+2))

Using the above formula, you have:

a = 1
b = (x+1)
c = (x+2)

Therefore, the final answer is:

(1*(x+2))/(x+1) = (x+2)/(x+1)

As for composite functions (f(g(x))), it's really just a matter of pure algebraic substitution.

Surely in your introductory algebra class in high school they made you put all variable substitutions in parantheses? Well, if they did, this is why.

To put it in perspective, treat f as an algebraic expression, and g as a singular unit:

f = 1/x
g = whatever

f(g) = 1/g

Clearly, g is equivalent to a set of parantheses. g must be evaluated before f can be. Therefore, when g is a large algebraic expression, you should always put parantheses around it to express this fact (and to enforce that you follow it when simplifying).