Q.Determine the range of the following expression $\dfrac{x^2+x+1}{x^2-x+1}$, $x \in \mathbb{R}$.

Range of a Quadratic Function

The range of a function is the complete set of values it can output. For a quadratic $f(x) = ax^2 + bx + c$ (with $a \neq 0$) the graph is a parabola with exactly one turning point, so the outputs run in one direction from that turning value. Finding the range is therefore the same as finding the vertex and asking, "does the curve go up from here, or down?"

The core idea

• If $a > 0$ the parabola opens upward. It has a lowest point (a minimum) and rises forever above it, so the range is $[\,y_{\min},\, \infty)$.
• If $a < 0$ it opens downward. It has a highest point (a maximum) and falls forever below it, so the range is $(-\infty,\, y_{\max}\,]$.

The turning point sits at the vertex, whose coordinates are

Why the vertex value is the boundary

Complete the square:

The squared bracket is never negative, so $f(x)$ can never go below (if $a>0$) or above (if $a<0$) the constant $c - \dfrac{b^2}{4a}$. That constant is exactly the extreme output — the edge of the range.

Worked feel

For $f(x) = 2x^2 - 8x + 5$: here $a = 2 > 0$, vertex at $x = 2$, value $f(2) = -3$. Range $= [-3, \infty)$. …