# How do you find the domain and range of g(x) = sqrt x / (2x^2+x+1)?

Feb 2, 2018

Domain: $\left[0 , + \infty\right)$ Range: $\left[0 , \approx 0.371\right]$

#### Explanation:

$g \left(x\right) = \frac{\sqrt{x}}{2 {x}^{2} + x + 1}$

Numerator: $\sqrt{x}$ is defined $\forall x \in \mathbb{R} : x \ge 0$

Denominator: $2 {x}^{2} + x + 1$

Discriminant of $2 {x}^{2} + x + 1 = {1}^{2} - 4 \times 2 \times 1 = - 7$

Since the discriminant $< 0 \to 2 {x}^{2} + x + 1 \ne 0$ for any $x \in \mathbb{R}$

Since the numerator is defined $\forall x \in \mathbb{R} : x \ge 0$ and the denominator is never 0 for real $x$ $g \left(x\right)$ is defined $\forall x \in \mathbb{R} : x \ge 0$

Hence, the domain of $g \left(x\right)$ is $\left[0 , + \infty\right)$

To find the range of $g \left(x\right)$ we need to find the upper and lower bounds.

By inspection, $g \left(0\right) = 0$

Since, $x \ge 0$ we can deduce that $g \left(x\right) \ge 0$

$\therefore {g}_{\min} = g \left(0\right) = 0$

Now consider ${\lim}_{x \to + \infty} g \left(x\right)$

$= {\lim}_{x \to + \infty} \frac{\sqrt{x}}{2 {x}^{2} + x + 1}$

$= {\lim}_{x \to + \infty} \frac{1}{\frac{2 {x}^{2} + x + 1}{x} ^ \left(\frac{1}{2}\right)}$

$= {\lim}_{x \to + \infty} \frac{1}{\left(2 {x}^{\frac{3}{2}} + {x}^{\frac{1}{2}} + {x}^{- \frac{1}{2}}\right)} = \frac{1}{\infty + 0} = 0$

NB: To find the finite upper bound of $g \left(x\right)$, if any, we would normally use calculus. However, since this question is in the Algebra section, I will solve this graphically.

Consider the graph of $g \left(x\right)$ below.
graph{ sqrtx/(2x^2+x+1) [-1.145, 3.18, -1.037, 1.126]}

We can observe from the graph that $g \left(x\right)$ has a maximum value of $\approx 0.371$ at $x = \frac{1}{3}$. We can further observe that $g \left(x\right)$ is declining from that point.

We have already shown that $g \left(x\right) \to 0$ as $x \to + \infty$
Hence we can deduce that ${f}_{\max} = f \left(\frac{1}{3}\right) \approx 0.371$ is the absolute maximum.

Hence, the range of $g \left(x\right)$ is $\left[0 , \approx 0.371\right]$