Monotonicity: Increasing and Decreasing Functions

Monotonicity describes whether a function goes up or down: a function is increasing on an interval if its values rise as x increases, and decreasing if they fall. In brief: f is increasing on an interval if x₁ < x₂ implies f(x₁) < f(x₂). For example, f(x) = 2x + 1 is increasing everywhere, while f(x) = x² decreases on (−∞, 0] and increases on [0, ∞).

Knowing where a function increases or decreases helps you sketch graphs, find maxima and minima and decide whether a function has an inverse. This guide gives the exact definitions, shows how to read monotonicity from graphs, tables and formulas and explains the idea of a turning point.

What do increasing and decreasing mean?

Let f be defined on an interval I. The function is strictly increasing on I if, for any x₁ < x₂ in I, we have f(x₁) < f(x₂). It is strictly decreasing if f(x₁) > f(x₂). If the inequalities are weak (≤ or ≥), the function is non-decreasing or non-increasing, and constant functions are allowed. Some textbooks use “increasing” for the weak version, so check which convention your course uses.

Monotonicity is always about an interval. A function such as x² is not increasing or decreasing on its whole domain, but it has a different behaviour on each side of 0. A function that is only increasing or only decreasing on its whole domain is called monotonic.

How do you read monotonicity from a graph?

Read the graph from left to right. Where the curve climbs, the function is increasing. Where it falls, the function is decreasing. Where it is horizontal, it is constant. Give the answer as x-intervals, not y-values.

Example

Suppose a graph rises from x = −3 to x = −1, falls from x = −1 to x = 2 and rises again after x = 2. Then it is increasing on (−3, −1) and (2, ∞) and decreasing on (−1, 2). The points at x = −1 and x = 2 are turning points. These are local extrema: a local maximum at −1 and a local minimum at 2.

How do you prove monotonicity with algebra?

1. Take two numbers x₁ < x₂ from the interval.
2. Compute f(x₂) − f(x₁) and factor it.
3. Determine the sign of the difference. A positive difference means increasing and a negative one means decreasing.

Example 1: f(x) = 3x − 5

f(x₂) − f(x₁) = 3(x₂ − x₁) > 0, because x₂ > x₁. So f is strictly increasing on all real numbers. In general, a line y = mx + b is increasing if m > 0 and decreasing if m < 0.

Example 2: f(x) = x² on [0, ∞)

f(x₂) − f(x₁) = x₂² − x₁² = (x₂ − x₁)(x₂ + x₁). On [0, ∞), both factors are positive (x₂ + x₁ > 0 since x₂ > x₁ ≥ 0). So f is increasing. For x₁ < x₂ ≤ 0 the second factor is negative, so f is decreasing there.

Example 3: f(x) = 1/x on (0, ∞)

f(x₂) − f(x₁) = 1/x₂ − 1/x₁ = (x₁ − x₂)/(x₁x₂) < 0. So f is decreasing on (0, ∞), and also on (−∞, 0), but not on the union of the two, because f(−1) = −1 is less than f(1) = 1.

Monotonicity of common functions

What are turning points and extrema?

A point where a function changes from increasing to decreasing is a local maximum, and where it changes from decreasing to increasing, a local minimum. For a parabola f(x) = a(x − h)² + k, the vertex (h, k) is the turning point: for a > 0 it is a minimum and the function decreases on (−∞, h] and increases on [h, ∞).

For f(x) = x² − 4x + 1 = (x − 2)² − 3, the turning point is (2, −3). The function is decreasing on (−∞, 2] and increasing on [2, ∞). In calculus, the sign of the derivative gives this information directly.

Where does the idea come from?

Ideas of growth and decline appear in ancient studies of motion, but the precise definition in terms of inequalities is a product of nineteenth-century analysis, which sought exact formulations of concepts such as continuity and limits. The link between monotonicity and the sign of the derivative was established through the mean value theorem of calculus.

Where is monotonicity used today?

A monotonic function is one-to-one, so it has an inverse. This matters for logarithms, which invert the increasing function 2^x. In economics, a demand function is typically decreasing in price. In statistics, a cumulative distribution function never decreases. In programming, searching a sorted list works because the order is monotonic.

Which mistakes are common?

Describing the y-values

Monotonicity is stated in x-intervals. Say “increasing on (2, ∞)”, not “increasing from 5 to 9”.

Calling a negative function decreasing

f(x) = −5 + 2x has negative outputs for small x but is increasing. Direction matters, not the sign of the values.

Joining intervals incorrectly

1/x decreases on each of its two branches, but not on the union, as shown above.

Ignoring endpoints

For x², the turning point x = 0 can be included in either interval, since the strict inequality still holds on [0, ∞).