Lecture Notes: Finding the Domain of Functions

Introduction

• Focus on finding domain for specific functions.
• Discussion on shifts of these functions.

Rational Functions

Reciprocal Function

• Function: ( f(x) = \frac{1}{x} )
• Vertical Asymptote: ( x = 0 )
• Horizontal Asymptote: ( y = 0 )
• Domain: All real numbers except ( x = 0 )
  • Interval notation: ((-\infty, 0) \cup (0, \infty))

Shifted Reciprocal Function

• Function: ( f(x) = \frac{1}{x+2} )
• Domain shifts as ( x + 2 \neq 0 ) implies ( x \neq -2 )
• Domain: All real numbers except ( x = -2 )
  • Interval notation: ((-\infty, -2) \cup (-2, \infty))

Root Functions

• Square roots of non-negative numbers are positive.
• Function: ( f(x) = \sqrt{x} )
  • Domain: ([0, \infty))

Shifted Root Function

• Function: ( f(x) = \sqrt{x-5} )
  • Domain: ([5, \infty))

Logarithmic Functions

• Function: ( f(x) = \ln(x) )
• Domain: ( x > 0 )
  • Interval notation: ((0, \infty))

Shifted Logarithmic Function

• Function: ( f(x) = \ln(1-x) )
  • Domain: ( x < 1 )
  • Interval notation: ((-\infty, 1))

Exponential Functions

• Function: ( f(x) = e^x )
  • Domain: All real numbers, ((-\infty, \infty))

Trigonometric Functions

Sine and Cosine Functions

• Domain: All real numbers, ((-\infty, \infty))
• Periodic functions with no domain restrictions.

Tangent Function

• Function: ( f(x) = \tan(x) = \frac{\sin(x)}{\cos(x)} )
• Domain: All real numbers except odd multiples of ( \frac{\pi}{2} )
  • Excluded values: Odd multiples of ( \frac{\pi}{2} )

Cotangent Function

• Function: ( f(x) = \cot(x) = \frac{\cos(x)}{\sin(x)} )
• Domain: All real numbers except integer multiples of ( \pi )
  • Excluded values: Integer multiples of ( \pi )

Secant and Cosecant

• Use reasoning similar to tangent and cotangent for finding domains as they are reciprocals of sine and cosine.