What Is the Reciprocal Function and Why Is It Self-Inverse?

The reciprocal function is $f(x)=\frac{1}{x}$, defined for $x\neq 0$. It is self-inverse because applying it twice returns the original input: $f(f(x))=x$. Its domain and range are both the non-zero real numbers.

Why Is It Self-Inverse?

An inverse function reverses the effect of the original function. To prove that $f$ is self-inverse, compose it with itself:

$$f(f(x))=f\left(\frac{1}{x}\right)=\frac{1}{\frac{1}{x}}=x.$$

Since $x\neq 0$, the value $\frac{1}{x}$ is also non-zero, so the second application is defined. The composition equals the identity function on the domain, proving that

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

The same conclusion can also be obtained algebraically by writing $y=\frac{1}{x}$, swapping $x$ and $y$ to get $$, and rearranging to . This method shows that the inverse has exactly the same rule. The domain restriction is essential: zero cannot be an input or output because division by zero is undefined.

For example, begin with $x=4$. The first application gives $f(4)=\frac{1}{4}$, and the second gives $f$. This also works for negative inputs: and .

Graphically, a function and its inverse are reflections in the line $y=x$. The graph of $y=\frac{1}{x}$ is unchanged by this reflection, confirming that it is self-inverse. Its branches lie in quadrants I and III and approach, but never meet, the coordinate axes.

Common Misconception

A common misconception is that $f^{-1}(x)$ always means $\frac{1}{f(x)}$. This is false. The notation denotes the inverse function, whereas means the reciprocal of the output. They coincide here only because this particular function is self-inverse.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches SL 2.8. It may be tested analytically on Paper 1 or graphically on Paper 2. To justify that the function is self-inverse, show $f(f(x))=x$, state the restriction $x\neq 0$, and explain that the domain and range match. When sketching, label both asymptotes and show the two correct branches. Do not confuse an inverse function with a reciprocal.