y = \dfrac{1}{x}

Graph of the Reciprocal Function $y = 1/x$

$y = \dfrac{1}{x}$ is the rational function that expresses inverse proportion¹. Multiplying both sides by $x$ gives $xy = 1$ , so the graph is a hyperbola whose asymptotes are the $x$ - and $y$ -axes.

Domain and range

The domain is $x \neq 0$
The range is $y \neq 0$
Odd function
Each branch decreases

No $x$ makes $\dfrac{1}{x} = 0$ , so the curve never touches the $x$ -axis.

Symmetry

Since $f(-x) = -f(x)$ , it is an odd function with point symmetry about the origin, and its graph splits between the first and third quadrants. It is also symmetric about the lines $y = x$ and $y = -x$ .

Asymptotes and limits

Approach	Behaviour
$x \to \pm\infty$	$y \to 0$
$x \to 0^{+}$	$y \to +\infty$
$x \to 0^{-}$	$y \to -\infty$

The $x$ -axis is a horizontal asymptote and the $y$ -axis a vertical one. The values leap from $-\infty$ to $+\infty$ across the origin, which is why the graph falls into two separate branches.

Monotonicity

The derivative is $y' = -\dfrac{1}{x^2}$ , negative for every $x \neq 0$ , so the function decreases on $x < 0$ and decreases on $x > 0$ . It is not decreasing on the real line as a whole: $x = -1$ gives $y = -1$ while $x = 1$ gives $y = 1$ , so the value rises as you cross the gap. It decreases only within a single branch.

Notable points

$x$	$y$
$-1$	$-1$
$\dfrac{1}{2}$	$2$
$1$	$1$
$2$	$\dfrac{1}{2}$

Every point on the curve satisfies $xy = 1$ , so the rectangle formed by dropping perpendiculars to the two axes always has area $1$ .

Relationships with other functions

The reciprocal function is its own inverse: solving $y = \dfrac{1}{x}$ for $x$ gives $x = \dfrac{1}{y}$ , so applying it twice returns the original value. Integrating it produces the natural logarithm.

\int \frac{dx}{x} = \ln|x| + C

Squaring the denominator instead gives $\dfrac{1}{x^2}$ , an even function whose values are always positive, with a markedly different shape.

Applications

Inverse proportion describes relationships in which a product stays constant.

Boyle's law, where pressure and volume vary inversely at fixed temperature²
Speed and the time needed to cover a fixed distance
Resistance and current at a fixed voltage
Sharing a fixed amount of work among a number of people

Multiplicative inverse, Wikipedia
Boyle's law, Wikipedia

Graph of the Reciprocal Function y=1/xy = 1/xy=1/x