How do you find the inverse to this function :/

this was on the test and i took forever on it and could not get it so i missed a bunch of the questions at the end of the test

ive tried everyting and cant seem to get it
f(x) = 3 -Sqrt(1-1x/2)

in words f of x is 3 minus the square root of 1 minus 1/2 x

if anybody can help thanks 10 points!

Let x = f(y) and then express y in terms of x:

x = 3 - sqrt(1 - 1y/2)

Subtract 3 from both sides:

x - 3 = -sqrt(1 - 1y/2)

Multiply both sides by negative one:

3 - x = sqrt(1 - 1y/2)

Square both sides:

(3 - x)² = 1 - 1y/2

Expand the left side using the F.O.I.L. method:

First terms

3•3 = 9

Outside terms:

3•-x = -3x

Inside terms:

-x•3 = -3x

Last terms:

-x•-x = x²

Collect the terms:

9 - 3x -3x + x² = 1 - 1y/2

Combine like terms:

9 - 6x + x² = 1 - 1y/2

Subtract 1 from both sides and flip the equation:

-1y/2 = 8 - 6x + x²

Multiply both sides of the equation by -2:

y = -16 + 12x - 2x² ANS

To check this is the inverse function (f^-1(x)) we should verify the property that the inverse function evaluated at the function yields x and viceversa:

f^-1(f(x)) = x

y = -16 + 12x - 2x²

Let x = f(x) = 3 - sqrt(1-1x/2)

y = -16 + 12{3 - sqrt(1-1x/2)} - 2{3 - sqrt(1-1x/2)}²

y = -16 + 36 - 12sqrt(1-1x/2) - 2{9 - 6sqrt(1-1x/2) + 1 - 1x/2}

y = 20 - 12sqrt(1-1x/2) - 18 + 12sqrt(1-1x/2) - 2 + x

y = x

This checks. You should also check that f(f^-1(x)) = x but I will leave that to you.

f(x) = 3 - sqrt(1 - x/2)

In order to find the inverse, first switch x and f(x):

x = 3 - sqrt(1 - f(x)/2)

Now, solve for f(x). First, we subtract 3 from each side:

x - 3 = -sqrt(1 - f(x)/2)

Next, multiply through by -1:

3 - x = sqrt(1 - f(x)/2)

Square both sides:

(3 - x)^2 = 1 - f(x)/2

Now, you don't NEED to, but I will expand the left term here for the sake of completeness:

9 - 6x + x^2 = 1 - f(x)/2

Rearrange to a more acceptable form:

x^2 - 6x + 9 = 1 - f(x)/2

Subtract 1 from each side:

x^2 - 6x + 8 = -f(x)/2

Multiply by -2:

-2x^2 + 12x - 16 = f(x)

So, the inverse function is:

f(x) = -2x^2 + 12x - 16