MODERATE EXERCISES

(2) There are important functions whose graphs are a part of some conic. For example, the graph of the square root function, s(x) = , is the upper half of the parabola represented by the equation y² = x. For each of these functions, find their domain, describe which part of what conic gives their graph and sketch their graph.

 (a) 

 (b) g(x) = 3 - 

 (c) h(x) = 

 (d) F(x) = 
 

 (a) 

      Domain :  16 - x2  0
                x2  16

                -4  x  4

     y2 = 16 - x2

     x2 + y2 = 16  which is a circle.

     It is the upper half of the circle given by x2 + y2 = 16

     


 (b) g(x) = 3 - 

     Domain : 3 - 4x - x2  0
              x2 + 4x - 3  0

              -2 -   x  -2 + 

      y = 3 - 

      y - 3 = -

      (y - 3)2 = 3 - 4x - x2
               = -(x2 + 4x - 3)
               = -(x + 2)2 + 7

      (x + 2)2 + (y - 3)2 = 7

      This is a circle.  
      g(x) represents the lower half of the circle given by the equation above.

     


 (c) h(x) = 

     domain :  1 + x2  0
               (-,)

     y2 = 1 + x2

     y2 - x2 = 1

     This is the equation of a hyperbola.
     Hence h(x) is the upper half of the hyperbola.

     


 (d) F(x) = 

     domain : x2 - 4  0
              x2  4

              x  -2 ,  x  2

      y2 = x2 - 4

      x2 - y2 = 4

      x2/4 - y2/4 = 1

      This is a hyperbola and F(x) is representing the upper half of this hyperbola.