Find the range of the function
as x varies over all real numbers.
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.
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.