Suppose that there are two functions, f and g.
The domain of f is the interval [-2,10] and the range of f is the interval (-15,2].
The domain of g is the interval (-5,7) and the range of g is the interval [-2,3).
Determine all information that you can about the domains and ranges of:
(a) b(x) = f(x)+g(x).
(b)$c(x) = f(x)( g(x)).
so, no effort on your part, eh?
To determine the domain and range of the functions b(x) = f(x) + g(x) and c(x) = f(x) * g(x), we need to consider the properties of addition and multiplication of functions and the given domains and ranges of f(x) and g(x).
(a) Domain and Range of b(x) = f(x) + g(x):
The domain of b(x) will be the intersection of the domains of f(x) and g(x), which means it will include all the values of x that are common to both f(x) and g(x). In this case, the domain of b(x) will be the intersection of [-2, 10] and (-5, 7), which results in the domain (-2, 7).
Now, let's determine the range of b(x). To do this, we need to consider the possible values that f(x) + g(x) can take. The range of b(x) will be the set of all values that f(x) + g(x) can produce.
For the given ranges of f(x) and g(x), we can determine the minimum possible value of f(x) + g(x) by taking the sum of the minimum values from each range, and the maximum possible value by taking the sum of the maximum values. Therefore, the range of b(x) will be the interval (-15 - 2, 2 + 3) = (-17, 5).
Therefore, the domain of b(x) = (-2, 7) and the range of b(x) = (-17, 5).
(b) Domain and Range of c(x) = f(x) * g(x):
The domain of c(x) will be the same as the domain of b(x), which is (-2, 7).
To find the range of c(x), we need to consider the values that f(x) * g(x) can produce. However, because we don't have information about the signs of the functions or the specific values within the given ranges, we cannot determine the precise range of c(x). We can only state that the range of c(x) will be a subset of the real numbers.
Therefore, the domain of c(x) = (-2, 7) and the range of c(x) is a subset of the real numbers.