SOLUTION: The function f is defined f(x) = x^2 - 6x + 2 for 0 ≤ x ≤ 3 a) state the range of f b) state the domain of f^-1 (x)

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