Find the range of the function

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

as x varies over all real numbers.

toodlestraws Aug 1, 2020

#1**+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.

amazingxin777 Aug 1, 2020

#2**+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) = ( -12x^{2} - 12x + 24 ) / ( (x^{2} + 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.

geno3141 Aug 1, 2020