Function library
Reciprocal Function
f(x) = 1/x
The hyperbola 1/x — two mirror branches divided by asymptotes on both axes.
The reciprocal function f(x) = 1/x is the graph of inverse proportion: double the input, halve the output. Its graph is a hyperbola with two branches — one in the first quadrant sweeping from +∞ down toward the x-axis, one in the third quadrant rising from −∞ up toward it — separated by an uncrossable gap at x = 0. The curve never touches either axis: the y-axis (x = 0) is a vertical asymptote and the x-axis (y = 0) is a horizontal one.
Wherever one quantity is divided by another, the reciprocal lurks: at fixed voltage, current is proportional to 1/R (Ohm’s law); at fixed amount of gas, pressure is proportional to 1/V (Boyle’s law); the time to complete a job is proportional to 1/(workers). The function is its own inverse — applying it twice returns x — and odd, with 180° rotational symmetry about the origin. Type 1/x into the calculator and zoom out: the branches flatten against the axes but never land.
Computed properties
Calculated by the graphing calculator's own math engine at build time for 1/x.
- Roots in [−10, 10]
- none found
- y-intercept f(0)
- undefined
- Local extrema in [−10, 10]
- none found
- Sample values
- f(-2) = -0.5 · f(-1) = -1 · f(0) = undefined · f(1) = 1 · f(2) = 0.5
- Derivative f′(1)
- -1.0000
- Integral ∫₀¹ f(x) dx
- undefined
Key facts
- Domain: all real x ≠ 0; range: all real y ≠ 0. Undefined at x = 0.
- Hyperbola with two branches: (0, ∞) gives positive values, (−∞, 0) gives negative values.
- Vertical asymptote x = 0; horizontal asymptote y = 0.
- Odd function: f(−x) = −f(x); 180° rotational symmetry about the origin; its own inverse.
- Passes through (1, 1) and (−1, −1); strictly decreasing on each branch.
- Derivative f′(x) = −1/x²; antiderivative is ln|x| + C.
What the reciprocal is
Taking a reciprocal means dividing 1 by the input: 1/2 = 0.5, 1/4 = 0.25, 1/0.5 = 2. Small inputs produce huge outputs and huge inputs produce tiny ones — the defining seesaw of inverse proportion. Because 1/(1/x) = x, the function is an involution: it undoes itself, so its graph is symmetric across the line y = x, the hallmark of inverse functions.
The function is odd, f(−x) = −f(x): the third-quadrant branch is the first-quadrant branch rotated 180° about the origin. Notable points are (1, 1) and (−1, −1), the only points where input equals output (solving 1/x = x gives x² = 1). Everywhere else, input and output differ — dramatically so near zero.
Two asymptotes, two branches
x = 0 is a vertical asymptote: as x → 0⁺ the values → +∞, as x → 0⁻ they → −∞, and 1/0 is undefined — division by zero has no meaning, so the branches can never join. y = 0 is a horizontal asymptote: as |x| → ∞ the values → 0, approaching the x-axis ever more closely without reaching it. The axes are walls the curve approaches but never touches.
Each branch is strictly decreasing: on (0, ∞), larger x gives smaller 1/x, and the same holds on (−∞, 0). But the function as a whole is not decreasing — it jumps from −∞ up to +∞ across the gap at zero. The derivative f′(x) = −1/x² is negative wherever defined, confirming the downhill slide on each branch, and the curve is convex on (0, ∞) and concave on (−∞, 0).
Where the reciprocal appears
Inverse proportionality is everywhere in science: Boyle’s law (P ∝ 1/V), Ohm’s law (I = V/R), the lens equation’s 1/f = 1/dₒ + 1/dᵢ, and gravitational and electrostatic forces falling as 1/r². In each case, doubling the denominator halves the result — the reciprocal’s signature. Frequency and period are reciprocals (f = 1/T): a 0.01 s period is a 100 Hz tone.
In calculus, 1/x is famous as the function whose antiderivative is not a power: ∫(1/x)dx = ln|x| + C, the integral that forced the invention of the logarithm. Its improper integral from 1 to ∞ diverges (the harmonic series’ continuous cousin), yet the same shape rotated gives Gabriel’s horn — finite volume, infinite surface area.
Frequently asked questions
Why is 1/0 undefined?
Division asks “what times the divisor gives the dividend?” — no number times 0 gives 1, so 1/0 has no answer. On the graph this shows up as the vertical asymptote: values blow up to ±∞ near zero but never settle on a value at zero itself.
Is 1/x increasing or decreasing?
Decreasing on each of its two intervals — pick any two positive numbers and the larger input gives the smaller output — but not decreasing overall, because it jumps from −∞ to +∞ across x = 0. The derivative −1/x² is negative everywhere the function is defined, which only speaks about behavior within each branch.
What is the integral of 1/x?
ln|x| + C. The power rule ∫xⁿ dx = x^(n+1)/(n+1) fails at n = −1 (division by zero), so 1/x needs its own antiderivative — historically, this integral is how the natural logarithm was first defined. The absolute value keeps the formula valid for negative x too.