domain of the function question

[bensmyname]

hey i have the following question
what is the domain of the function y=squareroot[(x+2)/(x-4)]?
its probably simple but i cant work it out the correct answer given was: X< or equal to -2, or x>4.
I thought that it wouldnt matter as long as x was not 4 because then y would be defined, but obviously not haha dont understand.

 

[Aladdin]

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\(\displaystyle f(x)\ \ is \ defined \ for \ \ \frac{x+2}{x-4} \ positive.\) And for the denominator to be different from zero.
(Corrected prof. Khan)

\(\displaystyle Set \ up \ the \ table \ diagram . \ . \ .\ .\)

 

[Deleted member 4993]

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\(\displaystyle f(x)\ \ is \ defined \ for \ \ \frac{x+2}{x-4} \ positive.\) <<< be careful at x = 4

\(\displaystyle Set \ up \ the \ table \ diagram . \ . \ .\ .\)

 

[BigGlenntheHeavy]

\(\displaystyle One \ way. \ Assume \ every \ function \ has \ a \ domain \ of \ (-\infty,\infty), \ barring \ no \ restrictions.\)

\(\displaystyle Now, \ f(x) \ = \ \frac{\sqrt{x+2}}{x-4}, \ does \ any \ restrictions \ apply?\)

\(\displaystyle Yes, \ first \ x+2 \ \ge \ 0 \ \implies \ x \ \ge \ -2 \ and \ x \ \ne \ 4, \ why?\)

\(\displaystyle Hence, \ the \ domain \ of \ f(x) \ is \ [-2,4)U(4,\infty)\)

\(\displaystyle See \ graph \ of \ f(x).\)

[attachment=0:2a1mpj3l]ccc.jpg[/attachment:2a1mpj3l]

 

[mmm4444bot]

Glenn, it looks like the radicand is (x + 2)/(x - 4).

Since the radicand cannot be negative, we solve the following inequality.

(x + 2)/(x - 4) ? 0

I would follow Aladdin's suggestion and make a sign chart (to organize the signs of the factors, in each of the three intervals).

The chart will show that the radicand is zero or more when x is -2 or smaller OR when x is bigger than 4.

I thought that it wouldnt matter as long as x was not 4 Try evaluating the given expression for x = 1, and see what happens.

 

[BigGlenntheHeavy]

\(\displaystyle Sorry, \ I \ misread \ the \ problem, \ however \ if \ f(x) \ = \ \sqrt\frac{x+2}{x-4}, \ then\)

\(\displaystyle \frac{x+2}{x-4} \ \ge \ 0 \ and \ x \ \ne \ 4, \ hence \ domain \ = \ (-\infty,-2]U(4,\infty), done \ by \ table.\)

\(\displaystyle See \ graph \ of \ f(x) \ below.\)

[attachment=0:mbci54vf]ddd.jpg[/attachment:mbci54vf]