Inverse Proportion

Article Contents

What is Inverse Proportion?
The Constant k
Table of Values
Graph: The Hyperbola
Comparing k > 0 and k < 0
Domain and Range
Asymptotes
Properties of Inverse Proportion
Real-Life Examples
Common Mistakes
Interactive Exercises

1. What is Inverse Proportion? {#what-is-inverse-proportion}

Inverse proportion is a function of the form:

f (x) = \frac{k}{x}, k \neq = 0

It describes the relationship "when $x$ doubles, $y$ halves." The product $x \cdot y = k$ is constant.

Key feature

As one variable gets larger, the other gets smaller -- they change in opposite ways.

2. The Constant k {#constant-k}

The number $k$ determines the shape and position of the graph.

k = x \cdot y = x \cdot f (x)

Finding k

If you know that when $x = 4$ , $y = 3$ :

k = 4 \cdot 3 = 12 ⟹ f (x) = \frac{12}{x}

3. Table of Values {#table-of-values}

Example: $f (x) = \frac{6}{x}$

$x$	$- 6$	$- 3$	$- 2$	$- 1$	$1$	$2$	$3$	$6$
$f (x)$	$- 1$	$- 2$	$- 3$	$- 6$	$6$	$3$	$2$	$1$

Notice:

We skip $x = 0$ (division by zero is undefined)
The product $x \cdot f (x) = 6$ is always constant
Positive $x$ gives positive $f (x)$ ; negative $x$ gives negative $f (x)$

4. Graph: The Hyperbola {#graph-hyperbola}

The graph of $f (x) = \frac{k}{x}$ is called a hyperbola. It has two separate branches.

The graph has two branches:

When $k > 0$ : one branch in quadrant I, one in quadrant III
The branches never touch the axes

5. Comparing k > 0 and k < 0 {#comparing-k}

	$k > 0$ (e.g. $y = \frac{2}{x}$ )	$k < 0$ (e.g. $y = \frac{- 2}{x}$ )
Branches in	Quadrants I and III	Quadrants II and IV
In quadrant I	Decreasing, positive	Does not exist
Behaviour	Both $x$ and $y$ have the same sign	$x$ and $y$ have opposite signs

6. Domain and Range {#domain-range}

Since we cannot divide by zero:

D (f) = R ∖ {0} = (- \infty, 0) \cup (0, + \infty)

The output $\frac{k}{x}$ can be any number except 0:

R (f) = R ∖ {0} = (- \infty, 0) \cup (0, + \infty)

See Domain of a Function and Range of a Function for more details.

7. Asymptotes {#asymptotes}

An asymptote is a line that the graph approaches but never touches.

The hyperbola $y = \frac{k}{x}$ has two asymptotes:

Asymptote	Equation	Meaning
Vertical	$x = 0$ (y-axis)	As $x \to 0$ , $	y	\to \infty$
Horizontal	$y = 0$ (x-axis)	As $	x	\to \infty $,$ y \to 0$

In words: the graph gets infinitely close to the axes but never reaches them.

8. Properties of Inverse Proportion {#properties}

Property	Value
Formula	$f (x) = \frac{k}{x}$ , $k \neq = 0$
Graph	Hyperbola (two branches)
Domain	$D (f) = R ∖ {0}$
Range	$R (f) = R ∖ {0}$
Asymptotes	$x = 0$ and $y = 0$
Key product	$x \cdot y = k$ is constant
Symmetry	The graph is symmetric about the origin

9. Real-Life Examples {#real-life-examples}

Workers and time

A job takes 120 person-hours. The time $t$ to finish depends on the number of workers $w$ :

t (w) = \frac{120}{w}

Workers $w$	$1$	$2$	$3$	$4$	$6$	$10$	$12$
Hours $t$	$120$	$60$	$40$	$30$	$20$	$12$	$10$

More workers means less time -- classic inverse proportion!

Speed and travel time

A journey is 240 km. The travel time $t$ depends on speed $v$ :

t (v) = \frac{240}{v}

Speed (km/h)	$40$	$60$	$80$	$120$
Time (h)	$6$	$4$	$3$	$2$

A pizza is cut into $n$ equal slices. Each person's share:

s (n) = \frac{1}{n}

As more people share, each gets less.

Pressure and volume (Boyle's Law)

At constant temperature, gas pressure $P$ and volume $V$ satisfy:

P \cdot V = k ⟹ P = \frac{k}{V}

10. Common Mistakes {#common-mistakes}

Mistake	Correction
Plotting at $x = 0$	$f (0)$ is undefined -- skip $x = 0$
Connecting the two branches	The branches are separate -- do not draw a line through the origin
Confusing with direct proportion	Direct: $y = k x$ (line). Inverse: $y = k / x$ (hyperbola)
Thinking the graph touches the axes	It approaches but never touches (asymptotes)

11. Interactive Exercises {#interactive-exercises}

Practice inverse proportion:

Summary

Concept	Description
Inverse proportion	$f (x) = \frac{k}{x}$ , $k \neq = 0$
Graph	Hyperbola with two branches
Key property	$x \cdot y = k$ is constant
Domain	$R ∖ {0}$
Range	$R ∖ {0}$
Asymptotes	$x = 0$ (vertical) and $y = 0$ (horizontal)

Introduction to Functions -- what is a function
Direct Proportion -- the other type of proportion
Domain of a Function -- domain restrictions
Functions and Graphs -- plotting and reading graphs
Summary of Functions -- quick reference card