Which of the Following Graphs Represent a One-One Function?
📝 Question
Determine which of the following graphs represents a one-one function.
(a) and (b) are given graphs.
✅ Solution
🔹 Step 1: Use Horizontal Line Test
A function is one-one if every horizontal line intersects the graph at most once.
🔹 Step 2: Analyze Graph (a)
In graph (a), a horizontal line cuts the curve at more than one point.
This means different values of \(x\) give the same value of \(y\).
Hence, graph (a) is not one-one.
🔹 Step 3: Analyze Graph (b)
In graph (b), every horizontal line cuts the curve only once.
Thus, each value of \(y\) corresponds to exactly one value of \(x\).
Hence, graph (b) is one-one.
🎯 Final Answer
\[ \boxed{\text{Graph (b) represents a one-one function, while graph (a) does not}} \]
🚀 Exam Shortcut
- Use horizontal line test
- Cut once ⇒ one-one function
- Cut more than once ⇒ not one-one
- Monotonic graphs ⇒ always one-one