Graphs of Functions and Their Inverses (Pure Mathematics 1 – 9709)

Core Concept

Why reflection in $y = x$?

If the point $(a, b)$ lies on the graph of $f$, then $f(a) = b$, which means $f^{-1}(b) = a$. So the point $(b, a)$ lies on the graph of $f^{-1}$.

The transformation that swaps the coordinates of every point — sending $(a, b) \mapsto (b, a)$ — is precisely a reflection in the line $y = x$.

This means:

• Every point on $f$ has a mirror image on $f^{-1}$ across $y = x$.
• The line $y = x$ itself is the axis of symmetry between the two curves.
• If a point lies on $y = x$ (i.e. $a = b$), it maps to itself — it is a fixed point of the reflection.

The one-one condition

Only a one-one (injective) function has an inverse that is itself a function. Graphically, a function is one-one if and only if every horizontal line intersects its graph at most once (the horizontal line test). If a function fails this test, its domain must be restricted to a suitable interval before an inverse can be drawn.

Domain and range swap

When $f$ has domain $A$ and range $B$:

• $f^{-1}$ has domain $B$ and range $A$.

On a graph, this swap is visible: the $x$-extent of the inverse curve equals the $y$-extent of the original, and vice versa.