12. Given $g(x)=-3\left(2^{x-1}\right)-5$, state
a) Initial value
b) the horizontal asymptote
c) the domain and range: $D=$
Range: \(\boxed{(-\infty, -\frac{13}{2})}\)
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: \(\boxed{(-\infty, -\frac{13}{2})}\)