Given the function f(x)=log8x , construct the function g(x) such that the graph of g(x) is the same as the graph of f(x) but shifted 7 units down, shrunk vertically by a factor of 1/5 , and reflected across the y-axis. In 1–2 sentences, describe the construction of the function and identify the correct function.

Answer 1

To construct the function \( g(x) \) from \( f(x) = \log_8(x) \), we first reflect the graph across the y-axis, giving us \( f(-x) = \log_8(-x) \). Then, we vertically shrink it by a factor of \( \frac{1}{5} \), resulting in \( \frac{1}{5} \log_8(-x) \), and finally, we shift the graph down by 7 units to obtain \( g(x) = \frac{1}{5} \log_8(-x) - 7 \).

Thus, the final function is: \[ g(x) = \frac{1}{5} \log_8(-x) - 7. \]