monotonically decreasing

Monotonically decreasing: compare outputs, not their signs

Direct answer: Here, monotonically decreasing means nonincreasing: for any x₁<x₂ in the interval, f(x₁)≥f(x₂). Equal outputs are allowed. Strict decrease requires f(x₁)>f(x₂) for every such pair. What matters is how the output changes as the input increases, not whether the output is negative.

How to prove a function is decreasing

1. Check the domain, then state one interval to investigate. Exclude points where the formula is undefined, such as a zero denominator.
2. Take any x₁<x₂ in the interval and calculate f(x₂)−f(x₁). Keep the subtraction in this order throughout the argument.
3. A difference that is always nonpositive proves the function is nonincreasing. A difference that is always negative proves it is strictly decreasing.
4. For a piecewise function, check pairs of inputs on different pieces as well as pairs within each piece. Two decreasing pieces can still have an upward jump between them.

Worked example: g(x)=8−3x on the real numbers

For any real x₁<x₂, subtract to obtain g(x₂)−g(x₁)=−3(x₂−x₁). Since x₂−x₁>0, this difference is negative. Every larger input therefore gives a smaller output. The table provides a few values to check against the proof.

The function is strictly decreasing over all real numbers. Every output in this table is positive, yet the outputs decrease as x increases. Decreasing does not mean negative.

Counterexamples and common mistakes

• Confusing strict and non-strict decrease. The constant function h(x)=5 is nonincreasing but not strictly decreasing; a flat interval is not strict decrease.
• Combining the two decreasing intervals of 1/x into a claim about its whole domain. It is strictly decreasing on each of (−∞,0) and (0,∞), but −1<1 while 1/(−1)=−1<1=1/1, so it is not decreasing across the whole domain.
• Seeing a minus sign and stopping the sign analysis. Other factors or a denominator can also be negative, so use the interval conditions before drawing a conclusion.
• Treating a few successful comparisons as a proof for every input. One counterexample can disprove a claim, but a few matching values cannot establish it.

Definition references and example notes

• Wolfram: nonincreasing and strictly decreasing functions
• OpenStax Precalculus 2e: increasing and decreasing intervals

Last reviewed: 2026-09-10. The references explain the definitions. The examples and counterexamples are AICSCA practice material, not official exam questions.