12. Given $g(x)=-3\left(2^{x-1}\right)-5$, state a) Initial value b) the horizontal asymptote c) the domain and range: $D=$

Step 1 ：Given the function \(g(x) = -3(2^{x-1}) - 5\)

Step 2 ：To find the initial value, evaluate \(g(0)\): \(g(0) = -3(2^{0-1}) - 5 = -3(2^{-1}) - 5 = -\frac{3}{2} - 5 = -\frac{13}{2}\)

Step 3 ：Initial value: \(\boxed{-\frac{13}{2}}\)

Step 4 ：As \(x\) approaches infinity, the term \(-3(2^{x-1})\) approaches \(-\infty\), so the horizontal asymptote is \(-\infty\)

Step 5 ：Horizontal asymptote: \(\boxed{-\infty}\)

Step 6 ：Domain (D): All real numbers

Step 7 ：Range: \(\boxed{(-\infty, -\frac{13}{2})}\)