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anonymous
 one year ago
Will METAL!!
Find the domain of the given function.
f(x) = square root of quantity x plus three divided by quantity x plus eight times quantity x minus two.
x > 0
All real numbers
x ≥ 3, x ≠ 2
x ≠ 8, x ≠ 3, x ≠ 2
anonymous
 one year ago
Will METAL!! Find the domain of the given function. f(x) = square root of quantity x plus three divided by quantity x plus eight times quantity x minus two. x > 0 All real numbers x ≥ 3, x ≠ 2 x ≠ 8, x ≠ 3, x ≠ 2

This Question is Closed

Directrix
 one year ago
Best ResponseYou've already chosen the best response.0Would you write this out in symbols? I am not following what this equation is: f(x) = square root of quantity x plus three divided by quantity x plus eight times quantity x minus two.

phi
 one year ago
Best ResponseYou've already chosen the best response.2Do you see the equation editor in the bottom left? that would be good to use

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\sqrt{x+3}/(x+8)(x2)\]

phi
 one year ago
Best ResponseYou've already chosen the best response.2meanwhile, the domain are all "legal" x values. In other words, all x values that do not cause trouble. What kinds of trouble? we don't want square roots of negative numbers. if an x value causes that, we don't allow x to be that value no divide by 0: if an x value causes us to divide by 0, we don't allow that. Use those rules to decide which x values are not allowed, and that tells you which are allowed.

phi
 one year ago
Best ResponseYou've already chosen the best response.2to answer, first look at \( \sqrt{x+3}\) any idea what x values might cause us to have the square root of a negative number ?

phi
 one year ago
Best ResponseYou've already chosen the best response.2negative means less than zero if x+3 < 0 then that means x+3 is negative. And we don't want that we want x+3 >=0 (that means x+3 is 0 or positive) add 3 to both sides , what do you get ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0if you add 3 to both sides then you get x>3?

phi
 one year ago
Best ResponseYou've already chosen the best response.2you mean x>=3 in other words, if we want x+3 to be zero or bigger we need x+3 >= 0 and that means x >= 3 (otherwise we would have a square root of a negative number, and we don't want that)

phi
 one year ago
Best ResponseYou've already chosen the best response.2so \( x\ge 3 \) is part of the answer. next, look that the "bottom" (x+8)(x2) what happens if x is 8 ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh I see it, wow your good.

phi
 one year ago
Best ResponseYou've already chosen the best response.2the bottom becomes (8+8)(82) and 8+8 is 0 0 times anything is 0 in other words the bottom becomes 0, and we would be dividing the top by 0 we don't allow divide by zero. so we can't let x be 8 luckily , x must be >= 3 (to keep the top legal), so we are ok (8 is smaller than 3, so we won't allow it) but we have (x+8)(x2) what happens if x= 2 ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.00 times anything = 0

phi
 one year ago
Best ResponseYou've already chosen the best response.2yes, and we have zero. we don't want zero in the bottom so we have to say x is not allowed to be 2 all together, the allowed x values are \[ x\ge3 , x\ne 2\] which means x is 3 or bigger, but not 2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Verify the identity. cos 4x + cos 2x = 2  2 sin^2 2x  2 sin^2 x

phi
 one year ago
Best ResponseYou've already chosen the best response.2It looks painful. Do you have the "halfangle" formula for cos 4x ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0isn't it =+square root 1+ cos theta/ 2

phi
 one year ago
Best ResponseYou've already chosen the best response.2there are 3 different versions \[ \cos(2a) = \cos^2(a)  \sin^2(a) = 2\cos^2(a)1 = 1  2 \sin^2(a) \]

phi
 one year ago
Best ResponseYou've already chosen the best response.2because in your problem you only have sin on the right side, I would use the last version \[ \cos(2a) = 1  2 \sin^2(a) \] can you use that to replace cos(4x) in your problem ?

phi
 one year ago
Best ResponseYou've already chosen the best response.2match up cos 4x with cos 2a write cos 4x as cos (2* 2x) and we match cos (2 * a) a is 2x in your problem

phi
 one year ago
Best ResponseYou've already chosen the best response.2\[ \cos(2a) = 1  2 \sin^2(a) \\ \cos(4x) = ? \]

phi
 one year ago
Best ResponseYou've already chosen the best response.2\[ \cos(2a) = 1  2 \sin^2(a) \\ \cos(4x) = 1  2 \sin^2(2x) \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so 12sin^2(2x) 12sin^2(2x) + cos 2x = 2  2 sin^2 2x  2 sin^2 x

phi
 one year ago
Best ResponseYou've already chosen the best response.2yes, and if we add +2sin^2(2x) to both sides, we can simplify that a bit

anonymous
 one year ago
Best ResponseYou've already chosen the best response.012sin^2(2x) + cos 2x = 2  2 sin^2 2x  2 sin^2 x +2sin^2x +2sin^2x  ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.012sin^2(2x) + cos 2x = 2  2 sin^2 2x  2 sin^2 x +2sin^2x +2sin^2x  1(2x) + cos 2x = 22 sin^2 (2x) I'm not sure

phi
 one year ago
Best ResponseYou've already chosen the best response.2oops, you should add 2 sin^2(2x) (not 2sin^2(x) )

phi
 one year ago
Best ResponseYou've already chosen the best response.212sin^2(2x) + cos 2x = 2  2 sin^2(2x)  2 sin^2 x +2sin^2(2x) +2 sin^2(2x)  1 + cos(2x) = 2  2 sin^2(x)

phi
 one year ago
Best ResponseYou've already chosen the best response.2It looks like we should use the same rule as before , but now on cos(2x)

phi
 one year ago
Best ResponseYou've already chosen the best response.2yes, put that in for cos(2x)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.01 + cos(2x) = 2  2 sin^2(x)  12sin^2(x)= 2sin^2(x)

phi
 one year ago
Best ResponseYou've already chosen the best response.2you replace cos(2x) with 12sin^2(x) like this 1 + cos(2x) = 2  2 sin^2(x) 1 + (12sin^2(2x)) = 2  2 sin^2(2x) now simplify the left side

phi
 one year ago
Best ResponseYou've already chosen the best response.2the left side is \[ 1 + 1  2\sin^2(2x) \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.01 + (12sin^2(2x)) = 2  2 sin^2(2x)  1+12sin^2(2x) = 22sin^2(2x)  22sin^2(2x) = 22sin^2(2x) yay!

phi
 one year ago
Best ResponseYou've already chosen the best response.2yes, both sides are obviously the same.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0can we do another problem, not like this one though.

phi
 one year ago
Best ResponseYou've already chosen the best response.2one more, but please make it a new post.
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