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

ChowMein Feb 15, 2019

#1**+4 **

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

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

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

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

.Rom Feb 15, 2019

#2**+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**+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**+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