Questions: Use transformations of the graph of (f(x)=4^x) to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graphs to determine the function's domain and range. (g(x)=4^-x) Determine the transformation(s) that is/are needed to go from (f(x)=4^x) to the given function. Select all that apply. A. shift 4 units downward B. reflect about the (y)-axis C. shift 4 units to the left D. shift 4 units upward E. shrink vertically F. shrink horizontally G. reflect about the (x)-axis H. stretch horizontally I. shift 4 units to the right J. stretch vertically

Solution Steps

The function \( g(x) = 4^{-x} \) can be obtained from \( f(x) = 4^x \) by reflecting about the \( y \)-axis. This is because replacing \( x \) with \(-x\) in the function \( f(x) = 4^x \) results in \( g(x) = 4^{-x} \).

The asymptote of the function \( f(x) = 4^x \) is the line \( y = 0 \). Since reflecting about the \( y \)-axis does not affect the horizontal asymptote, the asymptote of \( g(x) = 4^{-x} \) is also \( y = 0 \).

The domain of \( g(x) = 4^{-x} \) is all real numbers, \( (-\infty, \infty) \), because the function is defined for all \( x \). The range of \( g(x) = 4^{-x} \) is \( (0, \infty) \), as the function approaches 0 but never reaches it and increases without bound as \( x \) decreases.

Final Answer