Hi, I have this question, "Find the range of (2x^2)/(x^2+x+1)."
What is the best way to go about this? Should I graph it, or can I work it out algebraically?
HELP PLEASE...
Vocabulary point. This is a rational function, not an equation.
To graph it, you would first have to work it out algebraically. Graphing rational functions is not so simple as graphing lines.
Start by finding the y intercept algebraically.
Finding the y intercept is easy. What does the function equal when x = 0?
OK Now comes a slightly hard part. 2x^2 is obviously never negative.
So \(\displaystyle \dfrac{2x^2}{x^2 + x + 1}\) is never negative if \(\displaystyle x^2 + x + 1 > 0\ for\ all\ x.\)
Can you show that? How?
Now comes the really hard part. Does the function have a finite maximum?
Denis suggests the clever idea of saying it does and solving for it using the quadratic formula. He calls the maximum k.
\(\displaystyle \dfrac{2x^2}{x^2 + x + 1} = k \implies 2x^2 = kx^2 + kx + k \implies\)
\(\displaystyle (k - 2)x^2 + kx + k = 0 \implies x = \dfrac{- k \pm \sqrt{k^2 - 4(k - 2)k}}{2(k - 2)} = \dfrac{- k \pm \sqrt{k^2 - 4k^2 + 8k}}{2k - 4} = \dfrac{- k \pm \sqrt{8k - 3k^2}}{2k - 4}.\)
For x to be a real number, what can we say about k? Well obviously it must be a positive number. What else can we say about it?
That is a KILLER problem for first year algebra.