Question 25 (5 points)
Determine the inverse equation of the function $f(x)=2(x-3)^{2}+8$. Show all of your work. Complete your solution by hand and upload a picture of your solution.
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Step 1 ：Let's start by replacing the function notation f(x) with y, so we have y = 2*(x - 3)^2 + 8.

Step 2 ：Next, we swap the roles of x and y, meaning we replace every x in the equation with y and every y with x. This gives us x = 2*(y - 3)^2 + 8.

Step 3 ：Now, we solve the new equation for y. To do this, we first subtract 8 from both sides of the equation, which gives us x - 8 = 2*(y - 3)^2.

Step 4 ：Then, we divide both sides of the equation by 2, which gives us (x - 8)/2 = (y - 3)^2.

Step 5 ：Taking the square root of both sides of the equation, we get \(\sqrt{(x - 8)/2} = y - 3\).

Step 6 ：Finally, we add 3 to both sides of the equation to solve for y, which gives us \(y = \sqrt{(x - 8)/2} + 3\).

Step 7 ：However, since the original function is a quadratic, the inverse function will have two branches. Therefore, the inverse function is also \(y = -\sqrt{(x - 8)/2} + 3\).

Step 8 ：\(\boxed{f^{-1}(x) = \sqrt{(x - 8)/2} + 3, f^{-1}(x) = -\sqrt{(x - 8)/2} + 3}\)