Question

Is it true that ƒ −(g − h) = (ƒ − g) − 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 ƒ(x) = x2, g(x) = x, and h(x) = 1. Then (ƒ − (g − h))(x) = x2 − x + 1 but ((ƒ − g) − h)(x) = x2 − x − 1.

Answer 1

Answer: A

Explanation:

Ok, in the first expression we have:

f - (g - h)

this can be written as:

f + (-1)*(g - h)

Now we can distribute that -1, because this is a scalar.

Now we have:

f + (-1)*g - (-1)*h

This is equal to:

f - g + h = (f - g) + h.

So the relation: ƒ −(g − h) = (ƒ − g) − h is wrong, and this is because the subtraction operation dos not have the distributive property. (when we "break" the parentheses, the sign of h should change)