Find the inverse of each function. 
Is the inverse a function?
f(x) = (x-1)² + 3

First let's graph to see what function we are 
to find the inverse of.



Notice this is a function because it passes the
vertical line test. That is, NO point of it is
directly ABOVE any other point:

Substitute y for f(x)

y = (x-1)² + 3

Interchange x and y

x = (y-1)² + 3

Solve for y

x = y² - 2y + 1 + 3

x = y² - 2y + 4

Swap right side and left sides

y² - 2y + 4 = x

Get 0 on the right

y² - 2y + 4 - x = 0

For clarity put parentheses around the last two terms:

y² - 2y + (4-x) = 0

Solve by the quadratic formula:

Use the quadratic formula:
                  ______ 
            -b ± Öb²-4ac
        y = —————————————
                2a 

where a = 1; b = -2; c = (4-x)

                      _______________ 
             -(-2) ± Ö(-2)²-4(1)(4-x)
        y = ———————————————————————————
                      2(1) 

                  ________ 
             2 ± Ö4-4(4-x)
        y = ———————————————
                    2

                  _______ 
             2 ± Ö4-16+4x
        y = ——————————————
                   2

                  ______ 
             2 ± Ö-12+4x
        y = ——————————————
                   2
                  _____ 
             2 ± Ö4x-12
        y = —————————————
                  2

Now let's use the +

                  _____ 
             2 + Ö4x-12
        y = —————————————
                  2

                  ______ 
             2 + Ö4(x-3)
        y = ——————————————
                   2

                   ___ 
             2 + 2Öx-3
        y = ————————————
                  2

                     ___
             2     2Öx-3
        y = ——— ± ———————
             2       2
                  ___
         y = 1 ± Öx-3


Now let's use the + sign:

                  ___
         y = 1 + Öx-3

That's one half of the inverse.  Let's look at the
graph of that:



That is a function. That is, NO point of it is
directly ABOVE any other point.

Now let's use the - sign:

                  ___
         y = 1 - Öx-3

That's the other half of the inverse.  Let's look at the
graph of that:



That is also a function. That is, NO point of it is
directly ABOVE any other point.

Now let's put them both together:


 
That's the complete inverse. However, it is not a function,
because it does not pass the vertical line test. That is,
some of its points are directly ABOVE some of its other 
points. 

Let's put the original function on that same set of axes:



Now let's add the identity line, which has equation y = x



Now you can see that the inverse of the function is its 
reflection across this identity line, whose equation is
y = x, so called because the variables y and x are
"identically" equal in the equation y = x.

So f(x) = (x-1)² + 3 is a function, but its inverse is not
a function.

Edwin