Final Answer:
The value of (g - f)(-7) is -66. None of the option is correct.
Step-by-step explanation:
To solve this, we need to calculate the function (g - f)(x) and then evaluate it at x = -7.
Let's find the expression for (g - f)(x).
Given f(x) = x^2 - 4 and g(x) = 4x + 7, the expression (g - f)(x) is:
(g - f)(x) = g(x) - f(x) = (4x + 7) - (x^2 - 4).
Next, we simplify the expression:
(g - f)(x) = 4x + 7 - x^2 + 4 [Distributing the negative signs inside the parentheses]
(g - f)(x) = -x^2 + 4x + 7 + 4 [Reordering the terms to combine like terms]
(g - f)(x) = -x^2 + 4x + 11 [Combining constant terms 7 + 4]
Now we have the expression for (g - f)(x), which is:
(g - f)(x) = -x^2 + 4x + 11
Next, we'll evaluate this at x = -7:
(g - f)(-7) = -(-7)^2 + 4*(-7) + 11
(g - f)(-7) = -(49) - 28 + 11 [Performing the operations inside the parentheses]
(g - f)(-7) = -49 - 28 + 11 [Negating the 49]
(g - f)(-7) = -77 + 11 [Combining -49 and -28]
(g - f)(-7) = -66 [Combining -77 and 11]
So the value of (g - f)(-7) is -66, however, this is not one of the options provided, which means we need to check our calculation to see if there has been any error:
Let's check:
(g - f)(-7) = -(-7)^2 + 4*(-7) + 11
(g - f)(-7) = -49 - 28 + 11
(g - f)(-7) = -49 - 28 + 11
(g - f)(-7) = -77 + 11
(g - f)(-7) = -66
Re-evaluating, we come to the same result of -66. All steps have been checked and the arithmetic performed correctly. If the answer -66 is not provided among A, B, C, or D, it seems there has been an error in the options given.
To find the correct answer, (g - f)(-7) equals -66 which is not present in the provided options.