Graph of 1/x (the reciprocal function)

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = 1/x is a rectangular hyperbola: two separate branches, one in the first quadrant and one in the third, each approaching both axes without ever touching them.

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The graph of

y=1x

for −6 ≤ x ≤ 6. Dots mark nothing: it has no roots or turning points.

Key features

Domain: x ≠ 0
Range: y ≠ 0
Vertical asymptote: x = 0
Horizontal asymptote: y = 0
Monotonic: Decreasing on each branch (but not overall)
Symmetry: Odd; also symmetric in the lines y = x and y = −x
Notable points: (1, 1) and (−1, −1), the vertices of the hyperbola
y-intercept: None: x = 0 is an asymptote
Derivative: −1x²
Antiderivative: ln(|x|)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of 1/x looks like

As x grows, 1/x shrinks toward 0; as x approaches 0 from the right, 1/x grows without bound. The two axes are both asymptotes, and the branches are mirror images through the origin.

The function is its own inverse: if y = 1/x then x = 1/y. That is why the graph is symmetric in the line y = x. Every rectangle drawn from the origin to a point on the curve has area 1.

The derivative −1/x² is negative everywhere, so each branch decreases, yet 1/x is not a decreasing function overall: it jumps from −∞ to +∞ across the asymptote.

Points of interest

Found numerically to about twelve digits in the window −6 ≤ x ≤ 6, then shown in exact form where one exists.

x-intercepts (roots): None in this window
Local maxima: None in this window
Local minima: None in this window
Inflection points: None in this window

The graph has no roots, extrema or inflection points: its only features are its two asymptotes. The calculator breaks the curve at x = 0 instead of joining the branches.

Table of values

A few points to plot, with exact values where they are simple.

Values of 1/x
x	y = 1/x
−4	−1/4
−2	−1/2
−1	−1
−1/2	−2
1/2	2
1	1
2	1/2
4	1/4

How to graph 1/x by hand

1
Draw both axes as asymptotes (dashed if you like).
2
Plot (1, 1), (2, ½), (4, ¼) and (½, 2), (¼, 4) in the first quadrant.
3
Plot the same points with both signs flipped in the third quadrant.
4
Draw two smooth branches that hug the axes without touching them.

Transformations

How the graph moves when you change the formula. The original is 1/x in blue.

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y = 1/(x - 2) + 1 Asymptotes move to x = 2 and y = 1.
y = -1/x Reflected: branches in the second and fourth quadrants.
y = 4/x Further from the axes: the vertices move to (2, 2) and (−2, −2).

Common mistakes

Joining the two branches across x = 0.
Letting a branch cross an axis. 1/x is never 0, and it is undefined at x = 0.

Where it shows up

Inverse proportion: at a fixed distance, travel time is inversely proportional to speed.
Boyle’s law: at constant temperature, gas pressure is inversely proportional to volume.