Functions · step 11 of 13

Monotone & strictly increasing functions

f is increasing if x ≤ y implies f(x) ≤ f(y), and strictly increasing if x < y implies f(x) < f(y). The strict version forces injectivity immediately, which is why monotonicity is the fastest route to proving a function invertible. The definitional proof — take x < y and compare f(x) with f(y) — works even when f is not differentiable.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Take two arbitrary points with x < yEverything follows from comparing f(x) and f(y); this argument needs no calculus.
Compute the differenceShow f(y) − f(x) > 0 by factoring, using y − x > 0 as the driver.
Use the derivative when availablef′ > 0 on an interval implies strictly increasing there. The converse is not quite true — x³ is strictly increasing with f′(0) = 0.
Deduce injectivityStrictly monotone functions are injective: distinct inputs are ordered, so their images are strictly ordered and cannot be equal.

Worked example

Prove f(x) = x³ + x is strictly increasing on ℝ, and deduce it is injective.

Take x < y. Then f(y) − f(x) = (y³ − x³) + (y − x).
Factor: y³ − x³ = (y − x)(y² + xy + x²).
So f(y) − f(x) = (y − x)(y² + xy + x² + 1). The first factor is positive.
The second is positive too: y² + xy + x² = (y + x/2)² + 3x²/4 ≥ 0, so adding 1 makes it ≥ 1.

Answer. f(y) > f(x) whenever y > x, so f is strictly increasing and therefore injective.

Where marks get dropped

These are the specific errors that cost credit on monotone & strictly increasing functions questions — QED's rubric penalises each of them separately.

Confusing increasing with strictly increasing. A constant function is increasing in the non-strict sense but is not injective.
Assuming f′ > 0 is necessary for strict increase. x³ is strictly increasing on ℝ despite f′(0) = 0.
Checking a few sample points instead of proving the inequality for arbitrary x < y.

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Monotone & strictly increasing functions — frequently asked questions

Does monotone imply injective?

Strictly monotone does. Non-strict monotone does not — a function can be flat on an interval and still be increasing in the weak sense.

Is the inverse of an increasing function increasing?

Yes. If f is strictly increasing and invertible, f⁻¹ is strictly increasing too, which follows directly from the definition.

How does this relate to sequences?

A monotone bounded sequence converges — the monotone convergence theorem. Monotonicity is what replaces an explicit limit computation.

The rest of Functions

Injective, surjective, bijective, composition, inverse. Each subtopic below has its own method, worked example and mark-losing traps.

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