+0  
 
0
96
2
avatar+80 

Find the range of the function

 

f(x)= (x^2+14x+9)/(x^2+2x+3)


as x varies over all real numbers.

 Aug 1, 2020
 #1
avatar+549 
+2

My favorites! So we know that in a fraction, the denominator can never be 0. So x^2+2x+3 can not be equal to 0 and range is basically what numbers can be the output. Hopefully, that gets you started.

 Aug 1, 2020
 #2
avatar+21955 
+1

If you are in calculus:

 

1)  lim f(x)  as  x→infinity = 1

     lim f(x)  as  x→-infinity = 1

 

2)  find the first derivative (quotient rule): 

            f'(x)  =  ( -12x2 - 12x + 24 )  /  ( (x2 + 2x + 3)2 )

 

3)  set the first derivative equal to zero and solve:  x = -2     x = 1

 

4)  put these two values for x back into f(x) and find the y-values; these will give you the 

     maximum and minum y-values; therefore, you have the range.

 Aug 1, 2020

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