If the domain of a function F is the set of all real numbers and the domain of a function G is the set of all real numbers, under what circumstances do (F+G)(x) and (F/G)(x) have different domains?
Whenever G(x) = 0 then the domains would differ. For example, if F(x) =x+2 and G(x) = x-3 then (F+G)(x) = (x+2)+(x-3) the domain for (F+G)(x) would be all real numbers. (F/G)(x)= (x+2)/(x-3) would not exist when x=3 because you cannot divide a function by zero.
Whatever the domains of F(x) and G(x) are they would be the same for (F+G)(x)
but for (F/G)(x) you would have to restrict any values for which G(x) = 0
In effect you stated that with an example
If a graph of a quadratic function can have 0, 1 or 2 x-intercepts. How can you predict the number of x-intercepts without drawing the graph or (completely) solving the related equation?
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I really don't understand this, any elaboration will be greatly appreciated. thank in advance
You are correct. The domains of (F+G)(x) and (F/G)(x) will differ whenever G(x) = 0. In the given example, (F+G)(x) = (x+2)+(x-3) has a domain of all real numbers because there are no restrictions on addition. However, (F/G)(x) = (x+2)/(x-3) cannot be defined when x = 3 because division by zero is undefined. So, the domain of (F/G)(x) would exclude the value x = 3.
That's correct! The domains of (F+G)(x) and (F/G)(x) would differ whenever G(x) is equal to 0. In your example, when G(x) = x-3, dividing by zero would lead to an undefined value in (F/G)(x). Therefore, (F/G)(x) would not exist when x equals 3, resulting in a different domain compared to (F+G)(x).
To determine the domain of (F+G)(x), you simply consider the domain of both functions separately, which in this case is all real numbers. The addition of two functions with real number domains will also have a domain of all real numbers.
On the other hand, to determine the domain of (F/G)(x), you would need to exclude all values of x where G(x) is equal to 0 because division by zero is undefined. In your example, that would be x = 3. Thus, the domain of (F/G)(x) would be all real numbers except x = 3.