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Let the domain of the function f(x) be the interval (-4,4). What is the domain of the function \(f(\frac{x-2}{x+2})\)?

 Feb 15, 2019
 #1
avatar+4424 
+4

\(\text{we have 3 conditions} \\ -4 < \dfrac{x-2}{x+2} < 4 \text{ and } (x+2) \neq 0\\\)

 

\(\text{if }x+2>0 \text{ i.e. if }x > -2 \\ -4x-8 < x-2 < 4x+8\\ -6 < 5x \wedge -3x < 10\\ \dfrac{-6}{5} < x \wedge x > -\dfrac{10}{3}\\ \text{distilling all this we end up with simply }-\dfrac{6}{5} < x\)

 

\(\text{if }x+2< 0 \text{ i.e. if } x < -2\\ -8x-8 > x -2 > 4x+8\\ -6 > 9x \wedge -3x > 10\\ -\dfrac{2}{3} > x \wedge x < -\dfrac{10}{3}\\ \text{distiliing all this we end up with }x < -\dfrac{10}{3}\)

 

\(\text{Combining these results we get}\\ x \in \left(-\infty, -\dfrac{10}{3}\right) \cup \left(-\dfrac{6}{5}, \infty\right)\)

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 Feb 15, 2019
 #2
avatar+99282 
+3

Hi Rom,

I was trying to work out what your  \(\wedge \)    (\wedge)  meant ....

 

It intersection I think  ??       \(\cap\)              \cap   

 

I have never seen a wedge used before....

Melody  Feb 15, 2019
 #3
avatar+4424 
+2

it just means logical AND

 

I use it when sets aren't obviously involved but you need to meet 2 or more conditions.

Rom  Feb 15, 2019
 #4
avatar+99282 
+2

ok so you use    \cap = intersection    only for sets

           and      \wedge  for the same thing when it is not written in set notation.

 

But 

you use  \cup = union    for sets and non-sets notation...

 

Thanks  Rom.

Melody  Feb 15, 2019
 #5
avatar+4424 
+3

how closely I stick to those rules is inversely proportional to how many beers I've had... :D

Rom  Feb 15, 2019
 #6
avatar+99282 
+2

That is understandable.    wink

Melody  Feb 15, 2019
 #7
avatar+4424 
+2

oh.. yeah.. intervals get \cup too... intervals are like sets

Rom  Feb 15, 2019

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