Monotonicity Formula

$f(x) = x^2$ is NOT monotonic over all reals—it decreases for $x 0$.

The Formula

Increasing: a < b \Rightarrow f(a) < f(b); Decreasing: a < b \Rightarrow f(a) > f(b)

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

f(x) = 2x is monotonic increasing. g(x) = -x is monotonic decreasing.

Increasing: a < b \Rightarrow f(a) < f(b); decreasing: a < b \Rightarrow f(a) > f(b)

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.

f \text{ is monotone increasing} \iff \forall a, b \in D: a < b \Rightarrow f(a) \leq f(b); \; \text{strictly if } f(a) < f(b)

medium

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

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.

hard

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

Calling f(x) = x^2 monotonic — it decreases for x < 0 and increases for x > 0, so it changes direction
Confusing 'always positive' with 'always increasing' — f(x) = \frac{1}{x} is positive for x > 0 but decreasing
Thinking monotonic means the function never equals the same value twice — a constant function is technically non-decreasing

Monotonic functions have inverses and are easier to analyze.

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.

Increasing: a < b \Rightarrow f(a) < f(b); decreasing: a < b \Rightarrow f(a) > f(b)

Monotonic functions have inverses and are easier to analyze.

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

Before studying the Monotonicity formula, you should understand: function definition.