Please any one give full list of formulas from functions chapter for IIT preparation

Sure, let me walk you through a well-organized collection of the most important formulas and concepts from the Functions chapter, tailored specifically for IIT JEE preparation. This list will help you quickly revise key points and apply them confidently to problem-solving.

Basic Function Definitions

• Function: A function \( f \) from set \( A \) to set \( B \), written as \( f: A \rightarrow B \), assigns each element of \( A \) to exactly one element in \( B \).
• Domain: Set of all possible inputs \( x \).
• Codomain: Set \( B \) containing all potential outputs.
• Range: Actual set of values \( f(x) \) takes for \( x \in A \).

Common Types of Functions

• Identity function: \( f(x) = x \)
• Constant function: \( f(x) = c \)
• Linear function: \( f(x) = ax + b \)
• Quadratic function: \( f(x) = ax^2 + bx + c \)
• Modulus function: \( f(x) = |x| \)
• Greatest integer function: \( f(x) = \lfloor x \rfloor \)
• Signum function:
  • \( f(x) = 1 \) if \( x > 0 \)
  • \( f(x) = -1 \) if \( x < 0 \)
  • \( f(x) = 0 \) if \( x = 0 \)

Function Properties

• One-One (Injective): Different inputs produce different outputs.
• Onto (Surjective): Every element in codomain is mapped by some input.
• Bijective: Function is both injective and surjective.
• Even function: \( f(-x) = f(x) \) for all \( x \)
• Odd function: \( f(-x) = -f(x) \) for all \( x \)
• Periodic function: \( f(x + T) = f(x) \), smallest such \( T \) is the period.

Inverse and Composite Functions

• Composite function: \( (f \circ g)(x) = f(g(x)) \)
• Inverse function: \( f^{-1}(y) \) is defined such that \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \)

Domain and Range Tips

• For root function \( f(x) = \sqrt{x} \): Argument \( \geq 0 \)
• For log function \( f(x) = \log x \): Argument \( > 0 \)
• For rational function \( f(x) = \frac{1}{x} \): Denominator \( \ne 0 \)

Function Transformation Rules

• Vertical shift: \( f(x) + k \) shifts the graph up/down.
• Horizontal shift: \( f(x + k) \) shifts left/right.
• Vertical scaling: \( af(x) \), stretches or compresses vertically.
• Horizontal scaling: \( f(ax) \), stretches or compresses horizontally.
• Reflection:
  • \( -f(x) \): Reflects about x-axis
  • \( f(-x) \): Reflects about y-axis

Algebra of Functions

For functions \( f(x) \) and \( g(x) \):

• \( (f + g)(x) = f(x) + g(x) \)
• \( (f - g)(x) = f(x) - g(x) \)
• \( (fg)(x) = f(x) \cdot g(x) \)
• \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \), when \( g(x) \ne 0 \)

Special Functional Equations

• \( f(x+y) = f(x) + f(y) \) → Linear if f is continuous
• \( f(x+y) = f(x)f(y) \) → Exponential if f(0) = 1

Application Areas in JEE

• Modulus and greatest integer functions are frequent in graph-based problems.
• Inverse and composite functions show up often in algebraic equations.
• Transformations and symmetry are vital for sketching and interpreting graphs quickly.