What makes a rule dependable?

1 · Observe

Autos, fares and a surprise

An auto-rickshaw fare is ₹30 plus ₹14 per kilometre. Give the driver a distance and he gives you exactly one fare. That reliability is what mathematicians call a function. If the same distance could produce two different fares, you could not plan a budget.

2 · Express

Function notation

We write f(x) for the output of the rule f at input x. The set of allowed inputs is the domain. The set of outputs actually produced is the range.

3 · Visualize

Evaluating, and guarding the domain

Asking for a fare at −1 km is outside the domain. Code should say so loudly instead of returning a number. The same idea shows up in 1 ÷ (x − 3): any x works except 3, where the denominator is zero. Near 3 the output explodes, at x = 3.001 it is 1,000, and at 3 it does not exist.

4 · Discover

Not every rule is a function

The question 'what number squares to 4?' has two answers, +2 and −2. One input, two outputs: not a function. The square-root function fixes this by promising the non-negative answer, so √4 = 2 every time.

5 · Apply

Two worked examples

Example 1, a fare. fare(2.5) = 30 + 14 × 2.5 = 65. The same input always gives 65.

Example 2, a domain. g(x) = √(x − 1). For x = 5, g = 2. For x = −3 the square root of −4 is not a real number, so the domain is x ≥ 1.

6 · Where it goes wrong

Three ways it goes wrong

7 · In Python

A fare function with guards

Here is how to implement it, step by step. The code below does exactly these steps, and every line is editable.

1. Write the rule as a function with a name, such as fare(km).
2. State the domain: for fare, km must be zero or more.
3. Make the code refuse inputs outside the domain, loudly.
4. For a division, find the input that makes the denominator zero and exclude it.
5. Test the edges: just inside and just outside the domain.

import math

def fare(km):
    """Auto fare: rupees. Defined only for a distance of zero or more."""
    if km < 0:
        raise ValueError("distance cannot be negative")
    return 30 + 14 * km

for km in (0, 2.5, 10):
    print("fare(", km, ") =", fare(km))

try:
    fare(-1)
except ValueError as e:
    print("fare(-1) ->", e)

# a rule that is not a function of x alone: one input, two outputs
pairs = [(4, 2), (4, -2)]            # x = 4 gives both +2 and -2 as 'square roots'
print("inputs repeated with different outputs:", {x for x, _ in pairs if sum(1 for a, _ in pairs if a == x) > 1})
print("the function sqrt picks one:", math.sqrt(4))

# the domain of 1 / (x - 3): everything except 3
for x in (2, 3.001, 4):
    print("1 / (x - 3) at", x, "=", round(1 / (x - 3), 3))
try:
    1 / (3 - 3)
except ZeroDivisionError:
    print("1 / (3 - 3) -> undefined, 3 is outside the domain")

fare( 0 ) = 30
fare( 2.5 ) = 65.0
fare( 10 ) = 170
fare(-1) -> distance cannot be negative
inputs repeated with different outputs: {4}
the function sqrt picks one: 2.0
1 / (x - 3) at 2 = -1.0
1 / (x - 3) at 3.001 = 1000.0
1 / (x - 3) at 4 = 1.0
1 / (3 - 3) -> undefined, 3 is outside the domain