(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For what values of the constant c is the range of F(x) (-infinity,+infinity)

F(x) =g(x)/h(x)

g(x)=x^2 +2x +C

h(x)=x^2 + 4x + 3C

2. Relevant equations

3. The attempt at a solution

g(x)=(x+1)^2 +(C-1)

h(x)=(x+2)^2+(3C-4)

h(x)>g(x) for any x less then -c

and g(x) has roots at -2 +-(4-3C)^(1/2) and i think that h(-2 -(4-3C)^(1/2)) >0

so F(x) at (-2 -(4-3C)^(1/2) -a very small number) is +infinity

and F(x) at (-2 -(4-3C)^(1/2) +a very small number) is -infinity

so as long as h(x) has two real roots that are not the same then the range for f(x) is all reals

so my problem is to show it with some math and not a graphical way

and i think but cant prove very good that c can be any number less then 4/3, if anyone can help or point me in the right direction that would be great

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# Calculus(I) Problem. find the value of c so that the range is all reals

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