Suppose that f has a domain of A and g has a domain of B.
(a). What is the domain of f + g?
(b). What is the domain of fg?
(c). What is the domain of f/g?
Is there a mathematical rule or theorem for this type of question?
I know (a) and (b) is the intersection of domain A and B. But how would I express the domain of f/g?
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Yes, there is a mathematical rule for determining the domain of functions in algebra.
To find the domain of f + g, we need to consider the combined domain of both f and g. The domain of f + g is the set of all values that can be input into both f and g simultaneously. In other words, it is the intersection of the domains of f and g.
To find the domain of fg, we also need to consider the combined domain of f and g. However, since fg represents the product of f and g, the domain of fg is the same as the domain of both f and g.
To find the domain of f/g, we need to consider the combined domain of f and g again. However, since g appears in the denominator of the fraction, we need to ensure that g is not equal to zero in order to avoid division by zero. Therefore, the domain of f/g is the set of all values in the domain of f where g is not zero.
In summary:
(a) The domain of f + g is the intersection of the domains of f and g.
(b) The domain of fg is the same as the domain of f and g.
(c) The domain of f/g is the set of all values in the domain of f where g is not zero.
If there are any specific restrictions or conditions mentioned about the domains of f and g, those should also be taken into account when determining the domain of f + g, fg, and f/g.
Yes, there is a mathematical rule called the "domain of composite functions" that can be applied to answer these questions. To determine the domain of f + g, fg, and f/g, you need to consider the intersection of the domains of the individual functions involved.
(a) To find the domain of f + g, you need to consider the numbers for which both f and g are defined. In other words, it is the intersection of the domains of f and g, denoted as A ∩ B.
(b) To find the domain of fg (the product of f and g), you need to consider the numbers for which both f and g are defined. Again, this is the intersection of the domains of f and g: A ∩ B.
(c) To find the domain of f/g (the quotient of f and g), you need to consider the numbers for which both f and g are defined, excluding any values of g that would result in division by zero. Therefore, the domain of f/g is the intersection of the domains of f and (B \ {values that make g zero}), denoted as A ∩ (B \ {a value from B that makes g zero}).
Remember that in each case, the domain of the composite function is determined by the restrictions placed by the individual functions involved.
It's important to note that in order to apply this rule, you need to know the domains of f and g. If the domains are not explicitly defined, you might need additional information or context to determine them.