Can someone help me with showing the steps for getting a answer of [ 2,∞ ] for problem that says?

Question

The answer is [ 2,∞ ] The problem asks:
Given f(x) = x-3 and g(x) = √(x-2) Find the domain of the quotients function f/g?

Can someone show me the steps for getting the answer [ 2,∞ ] ?

Answer 1

See g(x) = √(x - 2)

Now √(x - 2) cannot be negative as then the function would become imaginary. The minimum value must be zero.

If x = 2, x - 2 = 0

Since any value of x lower than x = 2, makes x - 2 negative and in turn renders √(x - 2) imaginary, x cannot be any value lesser than 2. All values from 2 to positive infinity are applicable for g(x). It applies to f(x) too. Considering f(x), x can be anything, but in order for a set of values to satisfy both functions, [2,∞ ] is the solution.

Answer 2

The domain of the quotient is going to be the intersection of the domain of f, and the subset of the domain of g for which g(x) ≠ 0. Now the domain of f is all of R, the domain of g is [2, ∞) and g(x) = 0 at x = 2. So the domain of f/g is (2, ∞).

Note: it is not [2, ∞), much less [2, ∞] which doesn't even translate to a valid interval since ∞ is not a real number.

Answer 3

Ok here we go. I assume that you know that you have to turn that into a fraction. After that you have to multiply both top and bottom by the denominator because you cannot have a squareroot on the bottom of a fraction. When you do that you x-2 at the bottom. to find the domain you have to set the denominator equal to zero. Therefore you get x=2. I would not include the 2 because it would make the problem undefined. So I would write my answer as (2, infinity].

Answer 4

They probably want you to stay with real numbers, which would require that x-2 be nonnegative since you're taking the square root of that quantity. Therefore, x would have to be in the range [2, inf] to avoid imaginary numbers.

Answer 5

since f/g gives you (x - 3)/(sqrt(x - 2))

the dominator can't be 0 and you can't have a negative in the sqrt, the only possible values are greater than 2, so the answer would be (2,∞)

Answer 6

Domain of f/g will be:
►D(f/g) =[ Df Π Dg ] - { x | g(x)=0 }

so Df= R
Dg = [ 2 ,∞)
Df Π Dg - {x | g(x)=0 } = [ R Π [ 2 ,∞) ] - {2} =

► D(f/g) = ( 2, ∞)

hope this helps