8. Using your knowledge of transformations and inverse functions, and the parent function $f(x)=\sqrt{x}$, determine the domain and range of the inverse of the function $g(x)=13 \sqrt{-(x+20)}-41$. Explain your thought process. (3 marks)
Solution
Step 1 ：Swap x and y in the function g(x) = 13*sqrt(-x - 20) - 41 to find the inverse function. This gives us y = 13*sqrt(-x - 20) - 41.
Step 2 ：Solve the equation for x to get the inverse function. This gives us the inverse function x = -(y + 41)**2/169 - 20.
Step 3 ：The domain of the inverse function is the set of all real numbers for which the function is defined. This gives us the domain as \({-251/2 - 13\sqrt{253}/2, -251/2 + 13\sqrt{253}/2}\).
Step 4 ：The range of the inverse function is the set of all possible output values, which is the set of all real numbers that the function can take. This gives us the range as \({-(x + 41)^2/169 - 20}\).
Step 5 ：Final Answer: The domain of the inverse function is \(\boxed{{-251/2 - 13\sqrt{253}/2, -251/2 + 13\sqrt{253}/2}}\) and the range of the inverse function is \(\boxed{{-(x + 41)^2/169 - 20}}\).
