determine the range for:f(x)=9-8x-x²

To do it graphically, draw the graph by plotting points.



And you can see that the highest point is (-4,25)
and so the values of y are never less than 25, so the
range is (-oo, 25]

Replace f(x) by y

              y = 9 - 8x - x²

Solve for x

x² + 8x + y - 9 = 0

x² + 8x + (y-9) = 0

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

where a = 1; b = 8; c = (y-9)

                     ______________
             -(8) ± Ö(8)²-4(1)(y-9)
        x = ————————————————————————
                     2(1)
 
                   ________                 
             -8 ± Ö64-4y+36
        x = —————————————————
                    2

                   ______                
             -8 ± Ö100-4y
        x = ——————————————
                   2

                   ______                
             -8 ± Ö4(25-y)
        x = ——————————————
                   2
                    ____
             -8 ± 2Ö25-y
        x = ——————————————
                  2

                      ____
             -8     2Ö25-y
        x = ———— ± ——————————
              2        2
                  ____ 
        x = -4 ± Ö25-y

What's under the radical cannot be negative.

Therefore

25-y > 0

  -y > -25

   y < 25

Or in interval notation (-oo, 25]  

Edwin