Functions (Including Inverses)

Standard form

y = mx + c, where m is the gradient and c is the y-intercept. Its graph is a straight line, and it is the only one of the four whose gradient is the same everywhere.

Average gradient is just the gradient

Because the steepness never changes, the average gradient between any two points on a straight line is m itself. This is the only one of the four functions for which that is true, and it is worth saying out loud when a question asks.

Why this one matters for inverses

A non-horizontal straight line passes both the vertical and the horizontal line test, so its inverse is always a function — no restriction needed. Every other function in this list needs more care.

Three forms, three different gifts

y = a(x − p)² + q gives the turning point (p, q) and the axis of symmetry x = p.

y = a(x − x₁)(x − x₂) gives the x-intercepts directly.

y = ax² + bx + c gives the y-intercept c, and x = −b ÷ 2a gives the axis of symmetry.

Symmetry is the point

The parabola is symmetrical about x = p, so the two x-intercepts sit the same distance either side of it, and two different x-values give the same y. That symmetry is exactly why the inverse is not a function until you cut the graph in half.

The sign trap

In y = a(x − p)² + q the bracket already carries the minus, so y = (x − 1)² − 4 turns at (1, −4), not (−1, −4).

Standard form

y = ax − p + q, with a vertical asymptote at x = p and a horizontal asymptote at y = q. The graph has two separate branches and never touches either dashed line.

Its own kind of symmetry

The hyperbola has point symmetry about where the asymptotes cross, at (p, q) — rotate it half a turn about that point and it maps onto itself. There is no turning point, no maximum and no minimum.

The mistake that costs the marks

Joining the two branches through the vertical asymptote. They are separate curves — a pen stroke across x = p claims the function has a value there, and it has none.

Standard form

y = a·bˣ + q with b > 0 and b ≠ 1. The horizontal asymptote is y = q, and the graph approaches it on one side without ever reaching it.

Why its inverse needs no restriction

Because the graph is always increasing or always decreasing, no horizontal line can cut it twice. It passes the horizontal line test as it stands, so reflecting it in y = x gives a function straight away — the logarithmic function.

The mistake that costs the marks

Drawing the curve flattening onto the x-axis when the asymptote is somewhere else. If q = −4 the graph flattens towards y = −4, not towards zero — and the range is y > −4, not y > 0.

Why does this matter for inverses?

Only one-to-one functions have inverses that are also functions. If a function is many-to-one, its inverse fails the vertical line test and is therefore a relation, not a function — unless we restrict the domain.

Common misconception: f⁻¹(x) ≠ 1f(x)

The notation f⁻¹ means the inverse function (swapping inputs and outputs), not the reciprocal. For example, if f(x) = 2x, then f⁻¹(x) = x2, not 1(2x).

Key insight — points swap across y = x

If the point (a, b) lies on f, then the point (b, a) lies on f⁻¹. The line y = x is always the mirror line between a function and its inverse when both are drawn on the same set of axes.

Why the unrestricted parabola fails

y = x² maps both 2 and −2 to 4. Reflecting it gives a sideways parabola in which x = 4 has two y-values, +2 and −2 — so the reflection fails the vertical line test and is a relation, not a function.

The fix: throw half the parabola away

Restrict the domain to x ≥ 0 and the inverse is y = √x. Restrict it to x ≤ 0 instead and the inverse is y = −√x.

Either restriction works — but you must state which one you chose, because the two give different inverses.

Finding the inverse, step by step

Write y = ax², swap x and y to get x = ay², make y the subject to get y = ±√(x ÷ a), and then choose the sign that matches your restriction. The swap is the algebra of the reflection in y = x.

Where the marks are lost

Writing y = ±√x as the final answer. The ± is the honest result of the algebra, but it is exactly the thing that makes the answer not a function — the restriction is what lets you drop one sign, and dropping it without saying which restriction you used earns nothing.