Monotonicity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Monotonicity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.

Your age is monotonically increasing—it only goes up, never back down. A timer counting down is monotonically decreasing.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Monotonic means 'one direction only'—no turning back. Monotone functions are invertible over their full domain.

Common stuck point: f(x) = x^2 is NOT monotonic over all reals—it decreases for x < 0 then increases for x > 0.

Sense of Study hint: Pick three increasing x-values and compute f(x) for each -- if the outputs always go in one direction, it is monotonic.

Worked Examples

Example 1

medium
Is \(f(x) = 2x + 3\) monotonically increasing? Show that if \(x_1 < x_2\) then \(f(x_1) < f(x_2)\).

Solution

  1. 1
    Assume \(x_1 < x_2\).
  2. 2
    Multiply by 2 (positive, preserves inequality): \(2x_1 < 2x_2\).
  3. 3
    Add 3 to both sides: \(2x_1 + 3 < 2x_2 + 3\).
  4. 4
    So \(f(x_1) < f(x_2)\). ✓
  5. 5
    \(f\) is monotonically increasing on all of \(\mathbb{R}\).

Answer

Yes — \(f(x) = 2x+3\) is strictly increasing
A function is monotonically increasing when larger inputs always produce larger outputs. Here the positive slope (2) guarantees this.

Example 2

hard
Determine the intervals on which \(h(x) = x^3 - 3x\) is increasing and decreasing.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
For \(f(x) = -x + 5\), is it increasing or decreasing? Verify with two test values.

Example 2

hard
Show that \(f(x) = x^2\) is NOT monotone on all of \(\mathbb{R}\) by giving a counterexample, then state where it IS monotone.
← Back to Monotonicity Formula Explained Practice Mode

Related Concepts

FunctionInverse Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition