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The Reciprocal Function

Simple Explanation

The basic reciprocal function, y = 1/x, has two separate curved branches — one in the upper- right, one in the lower-left — that both approach the x-axis and y-axis but never touch them. Those two axes are called asymptotes.

Why Do We Need It?

Reciprocal functions model inverse-proportion relationships — when one quantity doubles, the other halves — a very common real-world pattern.

See It

Graph of y = 1/x

Formula

General Reciprocal Function

y = a / x (x ≠ 0)

A reciprocal function has a two-branch, hyperbola-like curve that approaches (but never touches) both axes — the axes are its asymptotes.

a: — a nonzero constant that scales the curve and determines which two quadrants it occupies
x ≠ 0: — x cannot be zero — the function is undefined there

When to use it: Whenever you need to recognize or sketch the shape of a reciprocal (inverse-proportion) function.

Worked Example

Evaluate a reciprocal function

For y = 4/x, find y when x = 8.

Why Does This Work?

As x grows larger, 1/x must shrink toward (but never reach) 0, since no finite x makes 1/x exactly 0 — this is exactly why the curve approaches, but never touches, the x-axis. Similarly, x itself can never be 0 (division by zero is undefined), which is why the curve never touches the y-axis either.

Real-Life Example

Speed and travel time for a fixed distance

For a fixed distance, travel time = distance/speed — as speed increases, travel time decreases proportionally.

This inverse relationship between speed and time follows exactly the reciprocal function shape: doubling speed halves the time.

Practice

For y = 6/x, find y when x = 3.

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Common mistake

Trying to evaluate the reciprocal function at x = 0 — it is undefined there, not equal to zero or any other value; the graph simply has a gap (asymptote) at x=0.

Quick Review

y = a/x has two branches and two asymptotes (the x-axis and y-axis).
The function is undefined at x = 0.
Models inverse-proportion relationships.