Cambridge O Level Additional Mathematics · Syllabus 4037 · Functions
Horizontal Line Test
What is Horizontal Line Test?
A graphical test for invertibility: a function has an inverse function over a stated domain exactly when every horizontal line meets its graph at most once. A horizontal line meeting the graph twice exhibits two different inputs sharing one output, which makes the function many-one and therefore not invertible on that domain.
This definition is part of the Functions chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Horizontal Line Test
Why does a many-one function have no inverse?
An inverse must send each output back to exactly one input. If two different inputs share an output, the inverse would have to return two values for that one input, which a function cannot do. Use the horizontal line test: a horizontal line meeting the graph more than once shows the function is many-one, so \(f(x)=x^{2}\) on \(\mathbb{R}\) has no inverse because \(f(2)=f(-2)=4\). Restricting the domain to \(x\ge 0\) makes it one-one and an inverse exists.