Answer:
- (f+g), (f-g), (f·g), domain is [-3, 3]
- f/g, domain is [-3, 3)
Explanation:
Given f(x) = √(3+x) and g(x) = √(3-x), you want the domains of various combinations of f and g.
Domains
The domain of each of these functions is the interval on which it is defined. The domain of the square root function is all argument values greater than or equal to zero.
Domain of f
The argument of the square root will be non-negative when ...
3 +x ≥ 0
x ≥ -3 . . . . . . subtract 3
-3 ≤ x . . . . . . written using left-pointing arrow
Domain of g
The argument of the square root will be non-negative when ...
3 -x ≥ 0
3 ≥ x . . . . . add x
x ≤ 3 . . . . . written using left-pointing arrow
Intersection of domains
Any combination of functions f and g will only be defined on the domain where both f and g are defined. That is, the domain will be the intersection of the domains of f and g:
(-3 ≤ x) ∩ (x ≤ 3) = -3 ≤ x ≤ 3
Domain of f+g
The domain of f+g is the intersection of the domains of f and g.
[-3, 3]
Domain of f-g
The domain of f-g is the intersection of the domains of f and g.
[-3, 3]
Domain of f·g
The domain of f ·g is the intersection of the domains of f and g.
[-3, 3]
Domain of f/g
The domain of f/g is the intersection of the domains of f and g, excluding the value of x that makes g(x) = 0. That value is x=3.
[-3, 3)
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Additional comment
The graphs of the combinations of functions are attached. You notice the graph of f/g tends to infinity as x → 3. It is undefined at x=3.