I think this discrepancy could be because of an extraneous root. The way that I am looking at this function, the value will always be negative, never zero (x-intercept) or positive. But if you tried solving for the value of x when f(x) is zero, I'm guessing that you went through the following steps:

0 = -3 minus the rest

3 =  negative of the rest

9 = the rest squared = (4*(2-x))/9

81 = 8-4x

73 = -4x

-73/4 = x

However, if you set x = -73/4 in the function above, f(x) = -6.

Set y equal to 0:

0 = -3 - (2root(2-x)) / 3

Add 3 on both sides:

3 = -(2root(2-x)) / 3

Multiply 3 on both sides:

9 = -2root(2-x)

Divide by -2 on both sides:

-9/2 = root(2-x)

The square root of a number cannot be negative, so there is no solution for y=0