Even though the graph of f(x) = x^2 - 6x + 2 is this:



since , we only use the part between and on the vertical lines 
at x=0 (the y-axis) and x=3, which is this:



To find the inverse of f(x) = x2 - 6x + 2  for 0 < x < 3

1. We replace f(x) by y,

, 

2. We replace x by y and replace y by x:

, 

We solve for y:

, 

Multiply through by -1 to make squared term positive:

, 

Enclose the term that represents "c" in parentheses:

, 

Use the quadratic formula:

 










Now we work out the domain 
Replace y by what y equals:

Add -3 to all three sides

Since it is between -3 and 0 we use the negative sign
for the square root. 

That tells us that for the inverse we use the negative
sign for the same square root in the inverse:

Going back to the inequality, we square all three sides. 
Squaring each side involves multiplying each side by 
itself, which is a negative quantity in the first two, 
and 0 is multiplied by 0, so we reverse the 
inequality signs:


Turn it around

Add -7 to all three sides:

That's the domain of the inverse function. So the inverse is

although we write f-1(x) for y:

  <---ANSWER

Here is the graph of the inverse on the same set of axes (in blue):



And you see that the inverse is the reflection of the original funcetion
across the identity line, whose equation is y = x  (where x and y are
identically equal and the identity line is the line that bisects the 1st and 3rd
quadrants (in green, dashed since it's not part of either graph).




Edwin

1. which has the interval notation [-7, 2]
2. which has the interval notation [-7, 2]