Cambridge O Level Additional Mathematics · Syllabus 4037 · Functions
Inverse Function
What is Inverse Function?
The inverse of a one-one function f, written f inverse, is the function that reverses the mapping: if f sends a to b then f inverse sends b back to a. It exists only when f is one-one over its stated domain, its domain is the range of f, and its range is the domain of f.
This definition is part of the Functions chapter in Cambridge O Level Additional Mathematics.
Common mistakes with Inverse Function
- “Every function has an inverse; I just have to find it.” Correct Only a one‑one function has an inverse function. \(f(x)=x^{2}\) on \(\mathbb{R}\) has none, because \(f(2)=f(-2)=4\). Restricting the domain to \(x\ge 0\) is what makes an inverse possible.
Questions students ask about Inverse Function
Does \(f^{-1}(x)\) mean \(\dfrac{1}{f(x)}\)?
No. The superscript \(-1\) in \(f^{-1}\) is not an index; it names the inverse function, the mapping that runs \(f\) backwards. If \(f(2)=7\) then \(f^{-1}(7)=2\), whereas \(\dfrac{1}{f(2)}=\dfrac{1}{7}\). In the same way \(f^{2}(x)\) means \(f(f(x))\), apply \(f\) twice, and not \([f(x)]^{2}\).
How do you find the domain of an inverse function?
The domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\). So after rearranging to find the rule for \(f^{-1}(x)\), go back to the original function, find its range, and state that as the domain of the inverse. The rule is only half the answer; an inverse without its domain is incomplete, and the domain is not free to choose.