How do you find intercepts, extrema, points of inflections, asymptotes and graph y=x/(x^2+1)?

Jan 4, 2017

graph{x/(x^2+1) [-10, 10, -1, 1]}

Explanation:

The domain of the function is the entire $\mathbb{R}$, as the denominator of the rational function is always $> 0$.

We have that:

${\lim}_{x \to - \infty} \frac{x}{{x}^{2} + 1} = 0$

${\lim}_{x \to + \infty} \frac{x}{{x}^{2} + 1} = 0$

We can see that $y \left(x\right)$ has the line $y = 0$ as horizontal asymptote on both sides. We can also see that:

$y \left(x\right) < 0$ for $x < 0$
$y \left(x\right) > 0$ for $x > 0$
$y \left(x\right) = 0$ for $x = 0$

So $x = 0$ is the only intercept.

$y ' \left(x\right) = \frac{1 - {x}^{2}}{{\left({x}^{2} + 1\right)}^{2}}$

As the denominator of $y ' \left(x\right)$ is always positive the function is differentiable everywhere and:

$y ' \left(x\right) < 0$ for $x \in \left(- \infty , - 1\right)$ and $\in \in \left(1 , + \infty\right)$
$y ' \left(x\right) > 0$ for $x \in \left(- 1 , 1\right)$
$y ' \left(x\right) = 0$ for $x = \pm 1$

Therefore $y \left(x\right)$ starts from $y = 0$ at $x \to - \infty$ and decreases until $x = - 1$ where it reaches a local minimum $y \left(- 1\right) = - \frac{1}{2}$. It then increases until $x = 1$ (changing sign at $x = 0$) where it reaches a local maximum at $y \left(1\right) = \frac{1}{2}$, and then decreases approaching zero indefinitely as $x \to + \infty$

$y ' ' \left(x\right) = \frac{2 x \left({x}^{2} - 3\right)}{{\left({x}^{2} + 1\right)}^{3}}$

so inflection points are: $x = 0$ and $x = \pm \sqrt{3}$ and the concavity is determined by the sign of $y ' ' \left(x\right)$:

for $x \in \left(- \infty , - \sqrt{3}\right) , y ' ' \left(x\right) < 0 , y \left(x\right)$ is concave down
for $x \in \left(- \sqrt{3} , 0\right) , y ' ' \left(x\right) > 0 , y \left(x\right)$ is concave up
for $x \in \left(0 , \sqrt{3}\right) , y ' ' \left(x\right) < 0 , y \left(x\right)$ is concave down
for $x \in \left(\sqrt{3} , + \infty\right) , y ' ' \left(x\right) > 0 , y \left(x\right)$ is concave up