Graphing rational functions ?Use the guidelines of this section to make a complete graph of f.

Solution:- Step1 Given function is 4x+4 f(x)= x +3 The zero of the denominator is x=± 3i so, the domain is { x; x= / ± 3i } This function consists of an odd function divided by an even function. The product of even and odd function is odd. Therefore, the graph is symmetric about the origin. Step2 Differentiate the given equation to find f’(x) we get, f’(x)= 4 - 2x(4x+4) x +3 (x +3) 2 2 4(x +3)8x 8x = (x +3) 2 4x 8x+12 = (x +3) 4(x +2x3) = 2 2 (x +3) Again differentiate f’(x) to find f’’(x) we get, 16x + 32x 48x 8x+8 f’’(x)= (x +3) - (x +3) 16x +32x 48x(8x+8)(x +3) = 2 3 (x +3) 8x +24x 72x24 = 3 (x +3) Step3 To get extreme values we have to use f’(x)=0 4(x +2x3) 2 =0 (x +3) 4(x +2x3) = 0 x +2x3=0 x +3x-x-3=0 x(x+3)-(x+3)=0 (x-1)(x+3)=0 x=1,-3 The critical points are 1,-3. Step4 To find inflection points we have to use f’’(x)=0 8x +24x 72x24 3 =0 (x +3) 8x +24x -72x-24=0 8(x +3x -9x-3)=0 x +3x -9x-3=0 If you solve this we get x=4.4...