Official Solution:The function \(g(x)\) is defined as the greatest integer less than or equal to \(x\), while the function \(h(x)\) is defined as the least integer greater than or equal to \(x\). What is the product \(g(1.7) \times h(2.3) \times g(-1.7) \times h(-2.3)\)?A. 6

B. 9

C. 12

D. 16

E. 24

In problems involving "decimal functions," which involve rounding decimals up or down to a nearby integer, we must be very careful to follow directions precisely. Here, we have two functions that have similar but distinct definitions.

To avoid confusion between the two functions, evaluate just one function's results first.

The function \(g(x)\) is defined as the greatest integer less than or equal to \(x\). So \(g(1.7) = 1\), while \(g(-1.7) = -2\). Notice how this function operates on negative numbers. The results are not symmetrical: \(g(-1.7)\) does not equal the negative of \(g(1.7)\).

Likewise, we have the function \(h(x)\) defined as the least integer greater than or equal to \(x\). So \(h(2.3) = 3\), while \(h(-2.3) = -2\). Again, the results are not symmetrical: \(h(-2.3)\) does not equal the negative of \(h(2.3)\).

Now we multiply the results together:

\(g(1.7) \times h(2.3) \times g(-1.7) \times h(-2.3) = 1 \times 3 \times (-2) \times (-2) = 12.\)

Answer: C.

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