Question

1-3 Function Operations and Composition

Is it true that (f - g - h) = (-9) - h? Explain why you believe the equation is true or provide a counterexample to show that it is not.
A. Yes; the subtraction operation has the Commutative Property.
B. Yes; the subtraction operation has the Associative Property.
C. No; the subtraction operation does not have the Distributive Property.
D. No; let f(x) = x², g(x) = x, and h(x) = 1. Then (f - g - h)(x) = x² - x + 1, but (f - 9) - h)(x) = x² - x - 1.

Answer 1

Option D is correct because subtraction is neither commutative nor associative, demonstrated by the counterexample where (f - g - h)(x) is not equal to f(x) - 9 - h(x).

Step-by-step explanation:

The correct answer is option D. The assertion that (f - g - h) = (-9) - h is not generally true because subtraction does not have either the Commutative or the Associative Property. This is illustrated with a counterexample: let f(x) = x2, g(x) = x, and h(x) = 1. If we calculate (f - g - h)(x) = x2 - x - 1 and compare it to what (-9) - h would imply, which is f(x) - 9 - h(x) = x2 - 9 - 1 = x2 - 10, we find that the two expressions are not equal. This demonstrates that subtraction is neither commutative nor associative, and thus (f - g - h) does not simplify to (-9) - h.

No; let f(x) = x², g(x) = x, and h(x) = 1. Then (f - g - h)(x) = x² - x + 1, but (f - 9 - h)(x) = x² - x - 1.

In this case, we can use specific functions to provide a counterexample to show that the equation is not true. By choosing f(x) = x², g(x) = x, and h(x) = 1, we can evaluate both sides of the equation and see that they are not equal.

Starting with (f - g - h)(x), we substitute the functions: (x² - x - 1). Now let's evaluate (f - 9 - h)(x): (x² - 9 - 1). Simplifying both expressions, we get x² - x + 1 and x² - x - 10 respectively. These are not equal, so the equation is not true.