Why Is the Inverse Function a Reflection in the Line y = x?

An inverse function reverses the input-output relationship of the original function. Reversing every ordered pair from ((a,b)) to ((b,a)) is exactly the geometric transformation produced by reflection in the line $y=x$.

Why the coordinates are swapped

Suppose $f(a)=b$. This means the point ((a,b)) lies on the graph of $y=f(x)$. By definition, the inverse reverses this operation, so

$$f^{-1}(b)=a.$$

Therefore, ((b,a)) lies on the graph of $y=f^{-1}(x)$. Reflection in $y=x$ maps every point ((a,b)) to ((b,a)), while points already on $y=x$ remain unchanged. Consequently, the graphs of a function and its inverse are mirror images across this line.

The line $y=x$ is the set of points whose coordinates are equal. A reflection across this line preserves each point’s perpendicular distance from the line but places it on the opposite side. Swapping coordinates does exactly this, so the coordinate rule and the geometric reflection describe the same transformation.

Algebraically, this is why finding an inverse involves exchanging $x$ and $y$, then solving for $y$.

For example, let

$$y=2x+3.$$

Exchange $x$ and $y$:

$$x=2y+3.$$

Solve for $y$:

$$y=\frac{x-3}{2}.$$

Hence,

$$f^{-1}(x)=\frac{x-3}{2}.$$

The point $(1,5)$ on $f$ corresponds to $(5,1)$ on $f^{-1}$, confirming the reflection.

This also explains why the domain of $f$ becomes the range of $f^{-1}$, and the range of $f$ becomes the domain of $f^{-1}$. In a graphing question, plot corresponding pairs to check that the two curves are symmetric about .

A function must be one-to-one to have an inverse that is also a function. If a horizontal line crosses the original graph more than once, its reflected graph fails the vertical-line test; the original domain must then be restricted.

A common misconception is that $f^{-1}(x)$ means $\frac{1}{f(x)}$. It does not: the inverse reverses a function, whereas the reciprocal divides by its output.

Exam technique

This is SL 2.2, Functions, domains, range, graphs and may appear on either GDC-active paper. Show the coordinate swap or exchange $x$ and $y$, state any required domain restriction, and remember that the domain and range are interchanged.