A decreasing function is a function whose output gets smaller as the input gets larger. If you pick any two points on the function where the second input is greater than the first, the second output will be less than or equal to the first. On a graph, this looks like a curve or line that falls from left to right.
The Formal Definition
A function f(x) is decreasing on an interval if, for any two values a and b in that interval where b > a, f(b) ≤ f(a). In plain terms: move to the right along the x-axis, and the y-values stay the same or drop.
There’s an important distinction between two types:
- Strictly decreasing: f(b) < f(a) whenever b > a. The output always drops, with no flat spots allowed.
- Non-strictly decreasing (also called non-increasing): f(b) ≤ f(a) whenever b > a. The output drops or stays the same, so flat segments are permitted.
Most textbooks and teachers mean “strictly decreasing” when they say “decreasing” without further qualification, but it’s worth checking context. When the word “monotonically” appears in front, as in “monotonically decreasing,” it means the function decreases across its entire domain or a specified interval, not just in a small neighborhood.
How It Looks on a Graph
A decreasing function slopes downward as you read the graph from left to right. Every tangent line you could draw along the curve during a decreasing interval has a negative slope. If the graph transitions from climbing upward to falling downward, the peak where that switch happens is a local maximum. Likewise, the bottom of a dip where the graph stops falling and starts rising is a local minimum.
A function doesn’t have to be decreasing everywhere. Many functions increase on some intervals and decrease on others. For example, a parabola opening upward like f(x) = x² decreases on the interval to the left of its vertex and increases to the right. When describing where a function decreases, you write the interval using x-values only. If a graph falls from the point (0, 4) to the point (2, -4), you’d say the function is decreasing on the interval (0, 2).
Using the Derivative to Identify Decreasing Intervals
If you know calculus, the derivative gives you a direct test. For a function f that is continuous on [a, b] and differentiable on (a, b):
- If f'(x) < 0 for every x in (a, b), then f is decreasing on [a, b].
- If f'(x) > 0, the function is increasing instead.
This makes sense intuitively. The derivative measures the slope of the function at each point. A negative slope means the curve is falling. So to find where a function decreases, you take the derivative, set it less than zero, and solve for x. The resulting intervals are where the function decreases.
For example, take f(x) = -x² + 6x. The derivative is f'(x) = -2x + 6. Setting f'(x) < 0 gives -2x + 6 < 0, which simplifies to x > 3. So the function is decreasing on the interval (3, ∞). For x < 3, the derivative is positive and the function is increasing.
Common Examples
Some familiar functions are decreasing over specific intervals or their entire domains:
- f(x) = -x: A straight line with a slope of -1, decreasing everywhere.
- f(x) = 1/x on (0, ∞): The curve drops as x grows, approaching zero but never reaching it.
- f(x) = cos(x) on [0, π]: Cosine starts at 1 when x = 0 and falls to -1 when x = π. Its derivative, -sin(x), is negative throughout that interval, confirming the decrease.
- Exponential decay, f(x) = e^(-x): Decreasing across its entire domain, falling toward zero as x increases.
Strictly Decreasing Functions and Inverses
One reason the strictly decreasing property matters is that it guarantees the function is one-to-one: every output corresponds to exactly one input. No two different x-values produce the same y-value. You can verify this visually with the horizontal line test. If every horizontal line crosses the graph at most once, the function is one-to-one.
A one-to-one function always has an inverse. So if f is strictly decreasing and continuous on an interval, its inverse function exists. The inverse of a strictly decreasing function is itself strictly decreasing. The cosine example above illustrates this nicely: because cos(x) is strictly decreasing on [0, π], it has an inverse on that interval, which is the arccosine function, arccos(x).
Concavity and Rate of Decrease
A function can be decreasing quickly or slowly, and concavity tells you how the rate of decrease is changing. If a decreasing function is concave down (curving like an upside-down bowl), the slope is becoming more negative, meaning the function is falling faster and faster. If it’s concave up (curving like a bowl), the slope is becoming less negative, meaning the function is still falling but slowing down, leveling off as it approaches a minimum or a horizontal asymptote.
You can check concavity with the second derivative. A negative second derivative means concave down; a positive second derivative means concave up. This distinction matters when you’re sketching curves or analyzing real-world models like cooling rates or population decline, where knowing whether a decrease is accelerating or tapering off changes the interpretation completely.