Textbook
On a stated domain, $f(x)\propto g(x)$ means that a constant $c$, independent of $x$, satisfies $f(x)=c\,g(x)$ for every admissible $x$. The constant may depend on other parameters held fixed during this comparison.
Where $g(x)\ne0$, the ratio $f(x)/g(x)$ must stay equal to $c$. Where $g(x)=0$, the equality also requires $f(x)=0$; division by zero is not a valid test. A single matching observation does not establish a relationship over an entire domain.
With a fixed unit price, cost is proportional to quantity. With an added fixed delivery charge, $f(x)=cx+b$ with $b\ne0$ is no longer directly proportional. Likewise, $x$ and $x^2$ both increase for $x>0$, but their ratio $x^2/x=x$ changes.
A fixed coefficient relates any two admissible points, not just adjacent observations. From $f(x)=c\,g(x)$ follows $f(x_1)g(x_2)=f(x_2)g(x_1)$, a comparison requiring no division. Agreement at several points, however, establishes only those points. A claim over the entire domain needs a formula or a justification of the model. Changing measurement units can change the numerical coefficient, so units must remain consistent.
Definitions and notation: Deep Learning: Notation.
In plain language
With a fixed price for identical pencils, doubling the quantity doubles the cost because every order uses the same multiplier. A fixed delivery fee breaks that rule by staying unchanged when the order doubles. A bulk discount also changes the multiplier, even with free delivery. “More means more expensive” does not establish proportionality; couriers and promotion departments can both disrupt an elegant formula.
Example
The constructed unit price is 3. The example compares a bill without delivery with one that adds a fixed fee of 5.
Proportionality: a constant ratio
One item
Without delivery, cost divided by quantity is 3.
Step 1 of 4: One item
Read all steps
One item
Without delivery, cost divided by quantity is 3.
- Quantity; cost; ratio
- 1current3included3included
Two items
Quantity and cost both double; the ratio stays fixed.
- Quantity; cost; ratio
- 2current6included3included
Delivery cost
One item now costs 8, so the ratio is 8.
- Quantity; cost; ratio
- 1current8excluded8excluded
Constancy check
For two items the ratio is 5.5: there is no constant multiplier.
- Quantity; cost; ratio
- 2current11excluded5.5excluded
After adding delivery, the totals 8 and 11 have ratios to quantity of 8 and 5.5. Their disagreement already disproves one coefficient for these orders. The number 3 still describes the pencils within each bill, but no longer describes the whole bill. A common error is to check only the variable cost and declare the total proportional despite its fixed additional charge.
A proportionality check requires a constant ratio and an explicit statement of which parameters remain fixed.