Monotonicity

Understanding Monotonicity

In the parent chapter on function analysis, you already saw what a function is and how to think about its input–output behavior. Monotonicity focuses on how a function behaves as the input moves left to right: does the function consistently go up, consistently go down, or change direction?

This chapter is about giving precise language and tools for describing those “always going up” or “always going down” behaviors.

Basic Ideas: Increasing and Decreasing

Informally:

• A function is increasing on an interval if, as you move to the right along the $x$–axis, the function values go up.
• A function is decreasing on an interval if, as you move to the right, the function values go down.

More precisely, let $f$ be a function and $I$ an interval (like $(1,5)$, $[0,\infty)$, or $(-\infty,2]$).

Increasing, Strictly Increasing

• $f$ is increasing on $I$ if for any $x_1,x_2 \in I$ with $x_1 < x_2$,
  $$
  f(x_1) \le f(x_2).
  $$
  The function never goes down; it may stay flat for a while.
• $f$ is strictly increasing on $I$ if for any $x_1,x_2 \in I$ with $x_1 < x_2$,
  $$
  f(x_1) < f(x_2).
  $$
  The function always goes up and never stays exactly the same at two different $x$-values.

So:

• “Increasing” allows flat segments (constant pieces).
• “Strictly increasing” does not.

Decreasing, Strictly Decreasing

• $f$ is decreasing on $I$ if for any $x_1,x_2 \in I$ with $x_1 < x_2$,
  $$
  f(x_1) \ge f(x_2).
  $$
  The function never goes up; it may stay flat.
• $f$ is strictly decreasing on $I$ if for any $x_1,x_2 \in I$ with $x_1 < x_2$,
  $$
  f(x_1) > f(x_2).
  $$

Again:

• “Decreasing” allows flat parts.
• “Strictly decreasing” does not.

Monotone Functions

A common umbrella term is:

• A function is monotone (or monotonic) on an interval if it is either increasing or decreasing on that interval (not necessarily strictly).
• It is strictly monotone if it is either strictly increasing or strictly decreasing.

So “monotone” means “always moving in one direction (or flat), never reversing direction.”

Examples of Monotonic Behavior

Here the focus is on describing the behavior, not on formal graphing techniques.

1. $f(x) = 3x + 1$ on all real numbers:
• If $x_1 < x_2$, then $3x_1 + 1 < 3x_2 + 1$.
• So $f$ is strictly increasing on $(-\infty,\infty)$.
2. $g(x) = -2x$ on all real numbers:
• If $x_1 < x_2$, then $-2x_1 > -2x_2$.
• So $g$ is strictly decreasing on $(-\infty,\infty)$.
3. $h(x) = 5$ (a constant function):
• For any $x_1 < x_2$, we have $h(x_1) = 5 = h(x_2)$.
• So $h$ is increasing and decreasing (because $5 \le 5$ and $5 \ge 5$), but not strictly either.
4. $k(x) = x^2$ on:
• $[0,\infty)$: if $0 \le x_1 < x_2$, then $x_1^2 < x_2^2$. So $k$ is strictly increasing on $[0,\infty)$.
• $(-\infty,0]$: if $x_1 < x_2 \le 0$, then $x_1^2 > x_2^2$. So $k$ is strictly decreasing on $(-\infty,0]$.
• $(-\infty,\infty)$: not monotone on the whole real line, because it first decreases (to the left of $0$) and then increases (to the right of $0$).

This last example illustrates an important point: a function can be monotone on some intervals but not on others.

Intervals of Monotonicity

For many functions, it is useful to identify where they are increasing or decreasing. Those are called their intervals of monotonicity.

For example, for $f(x) = x^3 - 3x$:

• The function goes down on some portion of the real line and up on another.
• The real line can often be split into subintervals such that on each one, the function is increasing or decreasing.

Later, in calculus, you will use derivatives to find these intervals. In precalculus, you can still reason about monotonicity by:

• Inspecting the graph.
• Thinking about the function’s formula and how its output changes when $x$ increases.

When describing intervals, always link them to “as $x$ increases within this interval, what happens to $f(x)$?”

Monotonicity and One-to-One Functions

In the chapter on inverse functions, you will learn about one-to-one functions (injective functions): functions that never take the same value twice at different $x$.

Monotonicity and being one-to-one are closely connected:

• A function that is strictly increasing on an interval is one-to-one on that interval.
• A function that is strictly decreasing on an interval is also one-to-one on that interval.

Reason: if $x_1 < x_2$ and $f$ is strictly increasing, then $f(x_1) < f(x_2)$, so $f(x_1) \neq f(x_2)$. That means no horizontal line cuts the graph more than once on that interval.

This is important because:

• Strict monotonicity on an interval guarantees that the function has an inverse on that interval.

You will use this idea in precalculus when restricting functions like $f(x) = x^2$ to $[0,\infty)$ so that they become one-to-one and hence invertible.

Recognizing Monotonicity from Graphs

Even without calculus, you should be comfortable reading monotonic behavior directly from a graph:

• To test whether $f$ is increasing on an interval $I$:
• Look only at the portion of the graph lying over $I$.
• As you move your finger along the curve from left to right:
• If you never move down (you may move up or stay level), the function is increasing on $I$.
• To test strictly increasing:
• Your finger must always move up (no flat segments) as you go left to right.
• For decreasing:
• Your finger never moves up as you go left to right on that interval.
• For strictly decreasing:
• Your finger always moves down as you go left to right.

Important details:

• Monotonicity is always discussed on an interval. A function can look “mostly increasing” but still not be called increasing on a large interval if it has even one small place where it turns around.

Piecewise Monotone Functions

Many real-world and exam-style functions are piecewise-defined: they use different formulas on different intervals. Such functions are often monotone on each piece.

For a piecewise function like
$$
f(x) =
\begin{cases}
2x + 1, & x \le 0, \\

• x + 4, & x > 0,
  \end{cases}
  $$
  you analyze monotonicity on each piece separately:
• On $(-\infty, 0]$, the formula is $2x + 1$, which is strictly increasing there.
• On $(0,\infty)$, the formula is $-x + 4$, which is strictly decreasing there.

On the entire real line, $f$ is not monotone because it switches from increasing to decreasing at $x = 0$.

Common Patterns

Some standard families of functions and their monotonicity:

• Linear functions $f(x) = mx + b$:
• If $m > 0$: strictly increasing on $(-\infty,\infty)$.
• If $m < 0$: strictly decreasing on $(-\infty,\infty)$.
• If $m = 0$: constant; increasing and decreasing (but not strictly).
• Power functions $f(x) = x^n$:
• For odd $n$ (e.g., $n = 1,3,5$): strictly increasing on $(-\infty,\infty)$.
• For even $n$ (e.g., $n = 2,4$): strictly decreasing on $(-\infty,0]$ and strictly increasing on $[0,\infty)$.
• Exponential functions $f(x) = a^x$ with $a > 0$, $a \neq 1$:
• If $a > 1$: strictly increasing on $(-\infty,\infty)$.
• If $0 < a < 1$: strictly decreasing on $(-\infty,\infty)$.

You do not need calculus to establish these patterns; they come from the algebraic properties of powers and exponentials that you study elsewhere.

Why Monotonicity Matters in Precalculus

Monotonicity plays several roles throughout precalculus and beyond:

• Understanding graphs: It tells you the overall “shape” and direction of a function on intervals.
• Finding inverses: Strict monotonicity on an interval ensures the function has an inverse there.
• Solving equations and inequalities: If you know $f$ is strictly increasing, then from $f(x_1) < f(x_2)$ you can deduce $x_1 < x_2$, and inequalities behave well when you apply $f$ to both sides.
• Describing behavior clearly: Instead of saying “the graph goes up here and down there,” you can say “$f$ is increasing on this interval and decreasing on that interval.”

As you move into calculus, monotonicity will be tightly connected to the sign of the derivative, but the basic notions and language you have learned here remain the same.