Foundations — Increasing and decreasing functions — intuitive definition

This page collects every symbol and idea the parent note Increasing and decreasing functions leans on, from absolute zero. Read it top to bottom; each item is built from the one above it.

1. Real numbers and the number line — where $x$ lives

Before we can say "$x_1$ is to the left of $x_2$", we need a place where "left" and "right" mean something.

Look at Figure s01. The two coloured dots mark two real numbers; the green and butter arrows show which direction counts as "bigger" and which as "smaller."

Why the topic needs it. The whole idea "bigger input → bigger output" only makes sense if "bigger" has a picture. On the number line, bigger simply means further right. Every comparison in this chapter is really a statement about who sits further right.

2. The symbols $<$ and $>$ — comparing two numbers

Why the topic needs it. Increasing is defined by "if $x_1$ is left of $x_2$, then $f(x_1)$ is below $f(x_2)$" — and $<$, $>$ are exactly the marks that record that left/right order. Without them, that sentence is just marks on paper.

3. Variables and the subscript notation $x_1,x_2$

Why the topic needs it. The definition talks about a pair of inputs and asks how their outputs compare. We need two distinct names for "the left point" and "the right point", so $x_1$ (left) and $x_2$ (right).

4. A function $f(x)$ — the machine that turns inputs into outputs

Look at Figure s02. The coral arrow carries an input $x$ into the machine $f$; the mint arrow carries the single output $f(x)$ back out. The little example shows $4$ going in and $11$ coming out.

• $f$ is the name of the machine.
• $x$ is what you feed it (the input).
• $f(x)$ is what comes out (the output).

So $f(x_1)$ means "feed the machine the input $x_1$, read the output." $f(x_2)$ means feed it $x_2$ instead.

Why the topic needs it. Increasing/decreasing is entirely about comparing two outputs, $f(x_1)$ versus $f(x_2)$. If you don't see $f(x)$ as "output for this input," the comparison has nothing to compare.

5. Ordered pairs and the coordinate plane — input on the floor, output as height

The number line handles one number. But a function ties an input to an output — two numbers at once. So we stand a second number line upright and cross it through zero.

First, how do we write two numbers together?

Look at Figure s03. Each coral dot is an ordered pair $(x,f(x))$; the dotted lines drop straight down to its input on the x-axis and straight left to its output height on the y-axis.

The key mental translation the whole chapter uses:

Why the topic needs it. Graphical analysis of monotonicity lives here — "uphill" and "downhill" are statements about dot heights on this plane.

6. Intervals and the symbol $I$ — where we're checking

A function can climb in one region and fall in another (like $x^2$). So we must always say over which stretch we're looking.

The symbol $\infty$ ("infinity") is not a number — it's shorthand for "keeps going with no right-hand end." A round bracket always hugs $\infty$ because you can never actually reach it. The symbol $\mathbb{R}$ is just a name for the collection of all real numbers.

Why the topic needs it. "$f$ is increasing on $I$" only has meaning once $I$ is fixed. This is Intervals and domain — the stage on which monotonicity is judged. It's also why the parent note keeps repeating "always specify the interval."

7. The logic symbols $\forall$, $\in$ and $⟹$

The formal definition reads $\forall x_1,x_2\in I:x_1<x_2⟹f(x_1)<f(x_2)$. Three new marks:

So the whole line says: "Take any two inputs in $I$. IF the first is left of the second, THEN its output is below the second's output."

Why the topic needs it. The $\forall$ is the reason checking a single pair fails. "For all" demands every pair obey the rule, not just a lucky one you tested. This is why worked examples switch to an algebraic argument (a general $x_1<x_2$) instead of specific numbers.

8. Putting it together — increasing AND decreasing, in one place

Now every symbol in the two definitions is earned. Here they are side by side.

Why does the inequality flip for decreasing? The input order stays $x_1<x_2$ (first still left of second). But "going downhill" means the right point is lower, so the output order reverses: $f(x_1)>f(x_2)$. Same $\forall$, same $⟹$ — only the final $<$ becomes $>$. This split into two mirror cases is the seed of Monotone functions.

9. Comparing outputs by subtraction — the difference $f(x_2)-f(x_1)$

The examples never eyeball "is this bigger?" — they subtract and check the sign. But first, one connecting symbol.

Why this tool and not another? Deciding which of two messy expressions is larger is hard by staring; deciding whether one expression is positive or negative is easy — you just track signs of its pieces. Subtraction turns a comparison into a single sign question. This is the seed of Inequality solving and later of Derivative and sign of f'(x), where the sign of a slope replaces the sign of a difference.

How these foundations feed the topic

Read it as: order (top-left) plus function machine (top-right) meet on the coordinate plane; the subtraction sign-check and the "for all" quantifier, restricted to an interval $I$, produce the definitions of increasing and decreasing.

Equipment checklist

Cover the right side and see if each rings true before continuing to the parent note.