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**H2 Mathematics - Functions**

A **function** is a relationship where every input of \(x\) has only one unique output \(y\).

• \(y = \ln x \Leftrightarrow e^y = x\)
• \(y = \cos x \Leftrightarrow \cos^{-1} y = x\)
• \(y = \sin x \Leftrightarrow \sin^{-1} y = x\)

**Types of Relations:**

**One-One (Function):**
• \(f^{-1}\) exists
• Inverse is also a function
• Passes Horizontal Line Test (HLT)

**One-Many:**
• NOT a function
• Example: \(x^2 = x + 5\), circle equations

**Many-One (Function):**
• \(f^{-1}\) doesn't exist
• \(f^{-1}\) would be one-to-many (not a function)
• Example: \(y = x^2\) for \(x \in \mathbb{R}\)

Conceptual Video

Conceptual Video

Video Solution:

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