Increasing and decreasing functions — intuitive definition: Visual walkthrough

Step 1 — Put two dots on the number line

WHAT. We choose two input numbers and call them $x_{1}$ and $x_{2}$ . That's it — two locations on the horizontal number line.

WHY. Increasing/decreasing is never about one point. A single dot has no "direction". You need two dots to ask "as I move from one to the other, did I go up or down?" Two is the smallest number of points that can show a trend.

WHAT IT LOOKS LIKE. Look at the figure. The horizontal line is the x-axis — the set of all possible inputs. The left dot is $x_{1}$ , the right dot is $x_{2}$ .

Figure — Increasing and decreasing functions — intuitive definition

Step 2 — Lift each dot up to the curve

WHAT. A function $f$ is a machine: feed it an input, it hands back an output. Feed it $x_{1}$ , out comes $f (x_{1})$ ; feed it $x_{2}$ , out comes $f (x_{2})$ . We draw those outputs as heights.

WHY. The number line only shows where we are (input). To see how high we climb we need a second axis — the vertical y-axis measuring output. Now each input dot rises to a point on the curve.

WHAT IT LOOKS LIKE. Each ground dot shoots a dashed line straight up until it hits the curve. The height where it lands is that input's output.

Step 3 — Compare the two heights (the whole game)

WHAT. We now ask the only question that matters: is the right dot higher or lower than the left dot? We measure this by subtracting the heights.

WHY subtract? Subtraction is the cleanest way to compare two numbers: the sign of $f (x_{2}) - f (x_{1})$ tells you everything.

If it's positive, the right height is bigger → we went up.
If it's negative, the right height is smaller → we went down.
If it's zero, same height → flat.

WHAT IT LOOKS LIKE. The vertical coloured bar between the two curve points is the difference. Magenta bar pointing up = positive; the arrow shows the climb from $f (x_{1})$ to $f (x_{2})$ .

Step 4 — Name the "up" case: increasing

WHAT. Suppose that for our chosen pair, $Δ > 0$ , i.e. $f (x_{1}) < f (x_{2})$ . The graph rose as we moved right.

WHY a special name? If this rising happens for every pair we could ever pick in the interval, the curve never dips — it's honestly "always climbing". We reserve the word increasing for exactly that.

WHAT IT LOOKS LIKE. Slide the pair anywhere along the interval — the right point is always higher. The green up-arrows never flip.

Step 5 — Name the "down" case: decreasing

WHAT. Now suppose $Δ < 0$ , i.e. $f (x_{1}) > f (x_{2})$ : the right point is lower. If this happens for every pair, the curve is decreasing.

WHY the inequality flips. The inputs still obey $x_{1} < x_{2}$ (that never changes — $x_{1}$ is on the left). But the heights now go the opposite way. Left-order becomes reversed height-order.

WHAT IT LOOKS LIKE. Same two dots, but the curve slopes down. The bar now points down (orange), the right point sits below the left.

Step 6 — The degenerate case: flat (constant)

WHAT. What if $Δ = 0$ for every pair? Then $f (x_{1}) = f (x_{2})$ always — the height never changes. This is a constant function like $f (x) = 5$ .

WHY it needs its own step. Beginners assume a curve must be either increasing or decreasing. Wrong. A flat line is neither strictly increasing nor strictly decreasing, because strict needs a genuine $<$ or $>$ , and $=$ is neither. It's the boundary case — the reason we say "strictly".

WHAT IT LOOKS LIKE. The two dots land at the same height. No bar, no arrow — a flat violet line.

Step 7 — The "changes its mind" case: $f (x) = x^{2}$

WHAT. Some functions are decreasing on one stretch and increasing on another. The classic is $f (x) = x^{2}$ : down on $(- \infty, 0)$ , up on $(0, \infty)$ .

WHY this proves monotonicity is local. Pick $x_{1} = - 2, x_{2} = - 1$ : heights $4$ then $1$ → down. Pick $x_{1} = 1, x_{2} = 2$ : heights $1$ then $4$ → up. Same function, opposite verdicts. So the answer depends entirely on which interval you're standing in.

WHAT IT LOOKS LIKE. The parabola falls into its valley at $x = 0$ , then climbs back out. Left half = orange (down), right half = green (up).

The one-picture summary

Everything compresses to one idea: carry a left→right pair of dots along the curve and read the sign of the height-change.

Height-change $+$ everywhere → increasing (green).
Height-change $-$ everywhere → decreasing (orange).
Height-change $0$ → flat (violet), strictly neither.
Sign changes → split the domain into intervals and label each.

Recall Feynman retelling (say it to a 12-year-old)

Put two fingers on a graph, left finger before the right finger. Now slide them along and watch which finger is higher. If the right finger is always higher, the curve is increasing — every step forward is a step up. If the right finger is always lower, it's decreasing — every step forward is a step down. If both fingers stay at the same height, it's flat, and we say it's strictly neither. And here's the twist: some curves make the right finger higher in one place and lower in another (like a valley) — so you must always say where you're looking. That "always higher" test, written in symbols, is exactly $x_{1} < x_{2} ⟹ f (x_{1}) < f (x_{2})$ .

Recall Quick self-test

Why do we need two points, not one? ::: One point has no direction; a trend (up/down) needs a pair to compare. What does the sign of $f (x_{2}) - f (x_{1})$ tell you? ::: $+$ increasing, $-$ decreasing, $0$ flat. Is $f (x) = 7$ strictly increasing? ::: No — it's flat, so strictly neither increasing nor decreasing. Why isn't checking one pair enough? ::: The definition uses $\forall$ (all pairs); $x^{2}$ passes $f (1) < f (2)$ yet fails on the left half.