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# Precalc- Function stuff

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

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

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

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

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Feb 15, 2019
#2
+4

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
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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
+3

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

Rom  Feb 15, 2019
#6
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That is understandable. Melody  Feb 15, 2019
#7
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oh.. yeah.. intervals get \cup too... intervals are like sets

Rom  Feb 15, 2019