A Hyperbola Branch $y = \sqrt{x^2+1}$

$y = \sqrt{x^2+1}$ places $x^2+1$ under a square root. Squaring gives $y^2 = x^2 + 1$, that is $y^2 - x^2 = 1$, the equation of a hyperbola. Since a square root is non-negative, this function represents the upper branch of that hyperbola.

Domain and range The radicand $x^2+1$ is always at least $1$, so the function is defined for all real $x$. Its smallest value is $\sqrt{1} = 1$, so the range is $y \ge 1$.

Symmetry Because $f(-x) = \sqrt{(-x)^2+1} = \sqrt{x^2+1} = f(x)$, it is even and its graph is symmetric about the $y$-axis.

Monotonicity and extremum The derivative is $f'(x) = \dfrac{x}{\sqrt{x^2+1}}$, negative for $x < 0$ and positive for $x > 0$. Hence the lowest point $(0, 1)$ is the minimum, and the curve increases monotonically on either side.

Asymptotes and limits For large $|x|$, $\sqrt{x^2+1} \approx |x|$, so the curve approaches the line $y = x$ on the right and $y = -x$ on the left; these are its two slant asymptotes. Indeed the derivative tends to $1$ as $x \to +\infty$ and to $-1$ as $x \to -\infty$.

Concavity The second derivative $f''(x) = \dfrac{1}{(x^2+1)^{3/2}}$ is always positive, so the graph is convex (concave up) everywhere and has no inflection points.

Relation to other functions Setting $x = \sinh t$ gives $\sqrt{x^2+1} = \sqrt{\sinh^2 t + 1} = \cosh t$, so the curve admits the parametrization $(\sinh t, \cosh t)$, reflecting the identity $\cosh^2 t - \sinh^2 t = 1$. Note that although it resembles the catenary $y = \cosh x$, it is a different curve.

Applications In relativity, the energy-momentum relation $E = \sqrt{(pc)^2 + (mc^2)^2}$ has exactly this form. The expression $\sqrt{x^2 + \varepsilon}$ is also used as a smooth approximation that rounds off the sharp corner of $|x|$, which is valuable in optimization.