Demonstrating a sense of variation through the definition

The rate-of-change method

To prove that a function is increasing or decreasing on an interval I, a reliable method is to examine the sign of the quotient: (f(b) - f(a)) / (b - a), for distinct a and b in I.

• If this quotient is always positive, f is increasing on I.
• If this quotient is always negative, f is decreasing on I.

This quotient is called the rate of increase: it measures the average change in f between a and b.

Detailed example

Let us show that f(x) = 2x + 5 is increasing on R. Let a and b be such that a is different from b.

f(b) - f(a) = (2b + 5) - (2a + 5) = 2b - 2a = 2(b - a)

Therefore, (f(b) - f(a)) / (b - a) = 2(b - a) / (b - a) = 2

The rate of increase is always 2, which is positive: f is therefore strictly increasing on ℝ. This makes sense, as an affine function of the form ax + b is increasing if a > 0 and decreasing if a < 0.

Example with a non-affine function

Let us show that f(x) = 1/x is decreasing on ]0; +∞[. For strictly positive and distinct a and b:

f(b) - f(a) = 1/b - 1/a = (a - b) / (ab)

(f(b) – f(a)) / (b – a) = (a – b) / ((ab)(b – a)) = –1 / (ab)

Since a > 0 and b > 0, ab > 0, so -1/(ab) is always negative: f is indeed decreasing on this interval.

A classic pitfall

It is essential to specify the domain on which we are working. For f(x) = 1/x, the reasoning above applies only to a and b of the same sign: on ]-∞; 0[ it is also decreasing, but we cannot compare a negative a and a positive b because 0 does not belong to the domain of f.