Decreasing Function — Definition, Graph & Examples
Decreasing Function
A function with a graph that moves downward as it is followed from left to right. For example, any line with a negative slope is decreasing.
Note: If a function is differentiable, then it is decreasing at all points where its derivative is negative.
See also
Key Formula
f′(x)<0⟹f is decreasing at x
Where:
- f(x) = The function being analyzed
- f′(x) = The derivative of f at x, representing the instantaneous rate of change
- x = An input value in the domain of f
Worked Example
Problem: Determine the interval(s) on which the function f(x) = −2x + 6 is decreasing.
Step 1: Find the derivative of f(x).
f′(x)=−2
Step 2: Check the sign of the derivative. Since f'(x) = −2, the derivative is negative for every value of x.
f′(x)=−2<0for all x
Step 3: Because the derivative is always negative, the function is decreasing on its entire domain.
f is decreasing on (−∞,∞)
Answer: f(x) = −2x + 6 is decreasing on (−∞, ∞). This makes sense because it is a line with a negative slope of −2.
Another Example
Unlike the first example (a straight line that is always decreasing), this polynomial is decreasing on only part of its domain. It shows how to use critical points and sign analysis to identify decreasing intervals.
Problem: Find the intervals on which f(x) = x³ − 12x is decreasing.
Step 1: Compute the derivative of f(x).
f′(x)=3x2−12
Step 2: Set the derivative equal to zero to find critical points.
3x2−12=0⟹x2=4⟹x=−2 or x=2
Step 3: Test a value in the interval (−2, 2). Choose x = 0.
f′(0)=3(0)2−12=−12<0
Step 4: Test a value outside this interval to confirm the function is not decreasing there. Choose x = 3.
f′(3)=3(9)−12=15>0
Step 5: Since f'(x) < 0 only on (−2, 2), the function is decreasing on that interval.
f is decreasing on (−2,2)
Answer: f(x) = x³ − 12x is decreasing on the interval (−2, 2).
Frequently Asked Questions
What is the difference between a decreasing function and a strictly decreasing function?
A decreasing (also called non-increasing) function satisfies f(a) ≥ f(b) whenever a < b — it can stay flat over some interval. A strictly decreasing function satisfies f(a) > f(b) whenever a < b, meaning the output must drop with every increase in input. In most algebra and calculus courses, 'decreasing' is used to mean strictly decreasing unless stated otherwise.
How do you tell if a function is decreasing from its graph?
Read the graph from left to right. If the curve or line falls (the y-values get smaller) over an interval, the function is decreasing on that interval. Visually, it looks like going downhill.
Can a function be both increasing and decreasing?
Not on the same interval, but a function can be increasing on one interval and decreasing on another. For instance, f(x) = x² is decreasing on (−∞, 0) and increasing on (0, ∞). The point where the behavior changes is typically a local maximum or minimum.
Decreasing Function vs. Increasing Function
| Decreasing Function | Increasing Function | |
|---|---|---|
| Definition | Output values fall as input values rise | Output values rise as input values rise |
| Derivative test | f'(x) < 0 on the interval | f'(x) > 0 on the interval |
| Slope of a line | Negative slope (m < 0) | Positive slope (m > 0) |
| Graph appearance | Falls from left to right | Rises from left to right |
| Example | f(x) = −3x + 5 | f(x) = 3x + 5 |
Why It Matters
Identifying where a function decreases is central to finding local maxima and minima in calculus — a function switches from increasing to decreasing at a local maximum. In real-world contexts, decreasing functions model situations like cooling temperatures, declining populations, or depreciating asset values. Understanding this concept also builds the foundation for curve sketching and optimization problems on standardized exams.
Common Mistakes
Mistake: Confusing a negative function value with a decreasing function. Students see f(x) < 0 and conclude the function is decreasing.
Correction: A function's sign (positive or negative output) is completely separate from whether it is increasing or decreasing. For example, f(x) = x − 10 is negative when x < 10, but it is increasing everywhere because its slope is positive. Check the derivative's sign, not the function's sign.
Mistake: Including endpoints of a closed interval when stating where a function is decreasing.
Correction: Intervals of increase or decrease are typically written as open intervals, e.g., (−2, 2), not [−2, 2]. At the critical points themselves, the derivative is zero, so the function is neither increasing nor decreasing at those exact points.
Related Terms
- Increasing Function — Opposite behavior — output rises as input rises
- Derivative — Used to test whether a function is decreasing
- Slope of a Line — Negative slope means a linear function is decreasing
- Function — The general concept that decreasing describes
- Differentiable — Required for the derivative test to apply
- Graph of an Equation or Inequality — Visual tool for identifying decreasing behavior
- Negative Number — Sign of the derivative or slope that signals decrease
- Line — Simplest example of a decreasing function when slope < 0