Applications and pitfalls to avoid

Inverse proportionality

Two quantities, x and y, are inversely proportional if y = k/x, where k is a non-zero constant: this is a variant of the inverse function.

Practical example: a journey of 120 km travelled at a speed v (in km/h) takes a time t = 120/v hours. If v doubles, t is halved: this is the typical behaviour of the function 1/x.

In science

• At constant electrical power, the current I = P/U is inversely proportional to the voltage U.
• In optics, the conjugation relation for lenses involves the inverses of distances.

Common pitfall no. 1: comparing reciprocals

To compare 1/a and 1/b, you must first check that a and b have the same sign. A common misconception is: “3 < 5, so 1/3 < 1/5”. In reality, 1/3 > 1/5 (i.e. approximately 0.33 > 0.20), because f is decreasing rather than increasing.

Common pitfall no. 2: forgetting x = 0

Many pupils forget to exclude x = 0 when solving an equation or analysing a function containing 1/x. You must always specify the domain of definition before starting any calculations.

Common pitfall no. 3: asymptotes

The curve approaches the axes but never touches them. Writing “f(x) = 0 when x is very large” is incorrect: one must state that f(x) tends towards 0 without ever reaching that value.

Key points

f(x) = 1/x models many real-world situations where one quantity decreases as the other increases, always within the domain ]∞; 0[ U ]0; +∞[.