Given two functions f and g, we want to see which statement is always true.
The only true statement is the third one:
"If a is in the domain of f + g. then I must be in the domain of f and it must be in the domain of g"
So what we can do, is try to find counterexamples for the given statements, if we can't, then the statements are true.
Note that the statements are really badly written, so I will complete them in a way that makes sense.
a) "If z is in the domain of f. then must be in the domain of f - g"
False, if z is not in the domain of g, then we can't define:
(f - g)(z)
This would happen if:
g = 1/(x - z)
b) "lf 2 is in the domain of f and 22 is in the domain of g, then the product, 222, must be in the domain of fg"
False, lets replace:
2 = z
22 = k
And let's say that:
f(x) = 1/(x - z*k)
g(x) = x
Then we have:
f*g(x) = x/(x - z*k)
Clearly, z*k (or 222 as you wrote it) is not on the domain.
c) "If a is in the domain of f + g. then I must be in the domain of f and it must be in the domain of g"
True, it only can be on the domain of the sum of the two functions if it belongs to the domain of each one.
d) "if a is not in the domain of (f + g) then a is not in the domain of g"
This is false, because a could be in the domain of g and not be in the domain of f.
Then the only true statement is the third one.
If you want to learn more, you can read:
https://brainly.com/question/4411112
