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) a(x) = [f(x)]^2.
(b) b(x) = f(x)+g(x).
(c)$c(x) = f(x)( g(x)).
iawdhoadhaoidhiawodwa
To determine the information about the domains and ranges of the composite functions, we need to consider the individual functions and their properties.
a) For the function a(x) = [f(x)]^2, we need to square the values of f(x).
Domain of a(x): The function f(x) has a domain [-2, 10]. Since squaring a real number does not affect the domain, the domain of a(x) will also be [-2, 10].
Range of a(x): The function f(x) has a range (-15, 2], and when we square a number within this range, the resulting values will only be positive. So, the range of a(x) will be the set of positive values in the range of f(x), which is (0, 4].
b) For the function b(x) = f(x) + g(x), we need to find the sum of the values of f(x) and g(x).
Domain of b(x): The function f(x) has a domain [-2, 10], and g(x) has a domain (-5, 7). The common region between the two domains is (-2, 7). Therefore, the domain of b(x) will be (-2, 7).
Range of b(x): The function f(x) has a range (-15, 2], and g(x) has a range [-2, 3). The sum of two numbers within these ranges will give us a range that includes the sum of the minimum and maximum values from the individual ranges. So, the range of b(x) will be (-15 + (-2), 2 + 3), which simplifies to (-17, 5).
c) For the function c(x) = f(x) * g(x), we need to multiply the values of f(x) and g(x).
Domain of c(x): The function f(x) has a domain [-2, 10], and g(x) has a domain (-5, 7). The common region between the two domains is (-2, 7). Therefore, the domain of c(x) will be (-2, 7).
Range of c(x): The function f(x) has a range (-15, 2], and g(x) has a range [-2, 3). The product of two numbers within these ranges will give us a range that includes the product of the minimum and maximum values from the individual ranges. So, the range of c(x) will be (-15*(-2), 2*3), which simplifies to (30, 6).
In summary:
(a) Domain of a(x): [-2, 10]
Range of a(x): (0, 4]
(b) Domain of b(x): (-2, 7)
Range of b(x): (-17, 5)
(c) Domain of c(x): (-2, 7)
Range of c(x): (6, 30)