Given f(x) = x-3 and g(x) = √(x-2) Can anyone find the domain of the quotients function f/g?

Question

Given f(x) = x-3 and g(x) = √(x-2)
Can anyone find the domain of the quotients function f/g?

Its one of those answers that’s like (#,-∞| or (#,-∞| U (#, #)
or whatever and I just can’t do these problems?

Answer 1

It just means find the allowable values of x. We usually do that by finding which ones are not allowed, and then saying the domain is all the others.
The two things to look for are fractions -- exclude any value that gives a zero denominator on a fraction; and square roots -- exclude any values that give a negative value under the radical sign.

In this case you have both:
√(x-2) is the denominator, so
x ≠ 2, and since also x-2 must not be negative, we find that in the domain
x>2
Some people would write that as
2 < x < ∞, but I don't like writing ∞ because so many people don't really understand it.
And it means the same as 2
So it's (?[ 2,∞ ]?) see, I don't remember which sort of brackets/parentheses means < and which is ≤, but I think the first answerer gave a response to that, and probably knows the convention.

Answer 2

ok well, if you want to find f/g, you divide the 2 functions:
(x-3) / sqrt(x-2) and it is simpler if the square root is in the top, so multiply the top and the bottom by sqrt(x-2). if you multiply the top and bottom by the same thing, you aren't changing anything (you're multiplying it by 1 essentially)

so (x-3)sqrt(x-2) / (x-2)

so now to find the domain, you have to see which values can x not be. you know that the square root can not be negative, so x can not be less than 2. you also know that the denominator can not be 0. so x can not equal 2.

every other value of x should be fine in this equation.

so the domain is x > 2. now to write it in that other form, you have to remember some stuff. the lower number always goes on the left (in the brackets) and the higher is on the right. a ( bracket means that the number is not included and a [ means that the number is included. when you have no end point, you have to use infinite or negative infinite and these brackets are ALWAYS (, not [.

So anyways, for this problem, the interval is (2,infinite). This means that x has to be greater than 2 and less than infinite. and x can not equal 2 (since the bracket is curly, not [ )

So here are some examples of the notation for the intervals

(-inf, inf) means that x can be anything
(2,5] means that x has to be greater than 2 and less than OR equal to 5

[2,3] means that x has to be between 2 and 3 and x is allowed to be equal to 2 or 3 too. so greater or equal to 2 and less than or equal to 3.

[0,inf] is incorrect notation, because a number can never be equal to infinite. [0, inf) is the correct notation which says that x has to be positive or 0.

(5,2) is also incorrect notation because the smaller number always has to be on the left. (2,5) is the correct notation which says x has to be between 2 and 5 but can not be equal to 2 or 5.

ummm,
(2,5)U(7,10) is basically combining 2 intervals. so x is between 2 and 5 or its between 7 and 10 and it is not equal to 2, 5, 7, or 10.
I think general notation for unions is that the left term is the lower interval.

Anyways, I hope you understand

Answer 3

hi, in case you attempt to simplify the equation you may persist with the persist with steps: 4x/5 -2x/3 first then upload the sum of that to x/2. So a million) discover an undemanding denominator, for this reason it particularly is 15. You write a clean fraction with a denominator of 15. Then divide 15 into the two denominators 5 and 3, then multiply those variety that are 3 and 5 into the numerators w/x so it would be 3 * 4x = 12x and 5 * 2x that's 10x. Then subtract the two numerators to get 2x/15 (the denominator continues to be consistent). Now take the 2d element, x/2, and upload it to 2x/15. You do this by way of looking yet another basic denominator which could be 30 then divide 30 into the two 15 and 2 that's 2 and 15. different the two those numbers into the numerators 2x and x and you will get 4x + 15x / 30. So your answer is 19x/30.

Answer 4

f(x)/g(x) = (x - 3)/√(x - 2)
If x ≤ 2 the denominator is 0 or imaginary, so your domain is 2 < x ≤ ∞, or simply x > 2

Answer 5

Since you can't divide by 0 or take the square root of a negative (in real numbers), x-2 must be > 0, so x must be in (2, ∞)

Answer 6

the only answer i am coming up with is [2, infinity) sorry I cant do the symbol for it. you can't have a negative under the radical no matter what.