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Increasing and Decreasing Functions — Definition, Formula & Examples

Increasing and decreasing functions describe the behavior of a function's output as the input moves from left to right. A function is increasing where its output rises as the input grows, and decreasing where its output falls as the input grows.

A function ff is increasing on an interval II if for all a,b∈Ia, b \in I with a<ba < b, it follows that f(a)<f(b)f(a) < f(b). It is decreasing on II if for all a,b∈Ia, b \in I with a<ba < b, it follows that f(a)>f(b)f(a) > f(b).

How It Works

To determine where a function increases or decreases, examine how f(x)f(x) changes as xx moves to the right. On a graph, increasing sections slope upward from left to right, while decreasing sections slope downward. In algebra or precalculus, you can test sample points in different intervals. In calculus, you use the first derivative: f′(x)>0f'(x) > 0 means increasing, and f′(x)<0f'(x) < 0 means decreasing. Intervals of increase and decrease are always stated in terms of the xx-values (input), not the yy-values.

Worked Example

Problem: Determine where the function f(x)=x2−4xf(x) = x^2 - 4x is increasing and where it is decreasing.
Find the vertex: Rewrite in vertex form or use x=−b/(2a)x = -b/(2a) to find the turning point.
x=−(−4)2(1)=2x = \frac{-(-4)}{2(1)} = 2
Evaluate at the vertex: Compute f(2)f(2) to confirm the minimum point.
f(2)=(2)2−4(2)=−4f(2) = (2)^2 - 4(2) = -4
State the intervals: Since the parabola opens upward (a=1>0a = 1 > 0), the function falls to the left of the vertex and rises to the right.
Decreasing on (−∞,2),Increasing on (2,∞)\text{Decreasing on } (-\infty, 2), \quad \text{Increasing on } (2, \infty)
Answer: f(x)=x2−4xf(x) = x^2 - 4x is decreasing on (−∞,2)(-\infty, 2) and increasing on (2,∞)(2, \infty).

Why It Matters

Identifying increasing and decreasing intervals is essential for sketching accurate graphs and finding maximum or minimum values. In AP Calculus, the first derivative test relies directly on this concept to classify critical points. In economics and physics, knowing where a function rises or falls helps model profit, velocity, and other real quantities.

Common Mistakes

Mistake: Writing intervals using yy-values instead of xx-values.
Correction: Increasing and decreasing intervals always refer to the input (xx) values. For example, say "increasing on (2,∞)(2, \infty)" rather than "increasing from −4-4 to ∞\infty."