Cambridge IGCSE Additional Mathematics · Syllabus 0606 · Functions
Function
What is Function?
A rule that assigns exactly one output to every input in a stated set called the domain. Two different inputs are allowed to share an output, but a single input may never produce two different outputs; a rule that does so is a relation rather than a function.
This definition is part of the Functions chapter in Cambridge IGCSE Additional Mathematics.
Function in context
A function is a rule that gives exactly one output for every input it is allowed to take. Everything else in this topic is a consequence of that one sentence: the domain is the set of inputs you are allowed to use, the range is the set of outputs actually produced, a composite feeds one function's output into another, and an inverse undoes the rule — which is only possible when no two inputs share an output.
Common mistakes with Function
- “\(f^{2}(x)\) means \([f(x)]^{2}\).” Correct \(f^{2}(x)=f(f(x))\) — apply \(f\) twice. For \(f(x)=2x+1\): \(f^{2}(3)=f(7)=15\), while \([f(3)]^{2}=7^{2}=49\). The syllabus does not use this iteration notation with trigonometric functions.
- “Composites can be read left to right, like English.” Correct In \(fg(x)\) the function nearest \(x\) acts first, so \(g\) goes first and \(f\) second. \(fg\) and \(gf\) are different functions in general.
- “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.
- “I found the rule for \(f^{-1}(x)\), so I know everything about it.” Correct The rule is half of the function; the domain is the other half, and it is not free to choose — the domain of \(f^{-1}\) is the range of \(f\). For \(f(x)=e^{2x}\) the rule \(f^{-1}(x)=\tfrac{1}{2}\ln x\) carries the domain \(x>0\). Find it every time, so that you can state it the moment a question asks.