For the given functions f(x) and g(x), find (f ∙ g)(x) or $ \left(\frac{f}{g}\right)(x) $ as indicated.
-$$f ( x ) = x ^ { 2 } - 4$$
A) Domain: $( - \infty , \infty )$
Range: $[ 4 , \infty )$

B) Domain: $[ - 4 , \infty )$
Range: $( - \infty , \infty )$

C) Domain: $( - \infty , \infty )$
Range: $[ - 4 , \infty )$

D) Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$

Question

For the given functions f(x) and g(x), find (f ∙ g)(x) or $ \left(\frac{f}{g}\right)(x) $ as indicated.
-$$f ( x ) = x ^ { 2 } - 4$$
A) Domain: $( - \infty , \infty )$
 Range: $[ 4 , \infty )$

B) Domain: $[ - 4 , \infty )$
Range: $( - \infty , \infty )$

C) Domain: $( - \infty , \infty )$
Range: $[ - 4 , \infty )$

D) Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$

Quizplus · Accepted Answer

To find (f ∙ g)(x) or $ \left(\frac{f}{g}\right)(x) $, we need to know the function g(x). Since it is not given, we cannot find the exact function. However, we can still determine the possible ranges for the composite function or quotient, based on the given range of f(x).

Since the range of f(x) is $[4, \infty)$, we know that all possible output values of f(x) are greater than or equal to 4. If we were to multiply or divide f(x) with another function g(x), the resulting range would still have a lower bound of 4.

Option A is not possible, since it suggests that the range of the composite function or quotient has no upper bound, which is unlikely given the properties of the given function f(x).

Option B is also not possible, since it suggests that the range of the composite function or quotient has no lower bound, which contradicts the given range of f(x).

Option C is the best choice, since it suggests that the range of the composite function or quotient is at least $[-4, \infty)$, which is consistent with the given range of f(x).

Option D is also possible, since it suggests that the range of the composite function or quotient has no specific bounds, which is technically true. However, option C is a more specific and useful answer in this case.

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