Start with the graph of the appropriate basic exponential function $f$ and use transformations to sketch the graph of the function $g$. State the domain and range of $g$ and the horizontal asymptote of its graph. \[ g(x)=20^{-x} \]

Step 1 ：The basic exponential function here is \(f(x) = a^x\), where \(a > 0\) and \(a \neq 1\). The function \(g(x) = 20^{-x}\) is a transformation of the basic exponential function where \(a = 20\) and the exponent is negated. This means the graph of \(g\) is a reflection of the graph of \(f\) in the y-axis.

Step 2 ：The domain of \(g\) is all real numbers because we can substitute any real number for \(x\) in the function \(g(x) = 20^{-x}\).

Step 3 ：The range of \(g\) is all positive real numbers because the base of the exponential function is greater than 1, so the function is always positive.

Step 4 ：The horizontal asymptote of the graph of \(g\) is the x-axis (y = 0) because as \(x\) approaches positive or negative infinity, \(20^{-x}\) approaches 0.

Step 5 ：The graph confirms our earlier thoughts. The function \(g(x) = 20^{-x}\) is a reflection of the basic exponential function \(f(x) = 20^x\) in the y-axis. The domain of \(g\) is all real numbers, the range of \(g\) is all positive real numbers, and the horizontal asymptote of the graph of \(g\) is the x-axis (y = 0).

Step 6 ：Final Answer: The domain of \(g\) is all real numbers, the range of \(g\) is all positive real numbers, and the horizontal asymptote of the graph of \(g\) is the x-axis (y = 0). In Latex format, the domain is \(\boxed{(-\infty, \infty)}\), the range is \(\boxed{(0, \infty)}\), and the horizontal asymptote is \(\boxed{y = 0}\).