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Unit 3.2 Properties of Real Functions . ‘Real function’ refers to a function whose domain and range are sets of real numbers. Categories of functions encountered in calculus and precalculus. Polynomial functions Rational functions Exponential functions Logarithmic functions

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unit 3 2 properties of real functions

Unit 3.2 Properties of Real Functions

‘Real function’ refers to a function whose domain and range are sets of real numbers.

categories of functions encountered in calculus and precalculus
Categories of functions encountered in calculus and precalculus
  • Polynomial functions
  • Rational functions
  • Exponential functions
  • Logarithmic functions
  • Trigonometric functions (and their inverses)
  • Sequences
analyzing real functions
Analyzing Real Functions
  • Typically done category by category
  • May miss some general principles used in analyzing all real functions
  • *In this unit we discuss properties of real functions that cross function category lines.
domains of real functions
Domains of Real Functions
  • The domain D of a real function f can be any subset of the real numbers R, but typically is one of two types:
  • (Type 1) A finite set of real numbers or a set of integers greater than or equal to a fixed integer k, where k is usually 0 or 1.
  • (Type 2) R itself or an interval in R, or a union of intervals in R.
the two types
The two types
  • Type 1: called discrete real functions. Includes sequences.
  • Type 2: called interval-based real functions. Includes the first 5 categories above.
characteristics to examine in analyzing a real function p 91
Characteristics to examine in analyzing a real function (p.91)
  • Domain: Is f discrete? Interval based?
  • Singularities and asymptotes: Where is f undefined? Does it have vertical asymptotes?
  • Range: What are the possible values of f?
  • Zeros: Where does f intersect the x-axis?
  • Maxima (minima), Relative maxima (minima): Find the greatest or least value of f (or f on some interval)
characteristics cont
Characteristics (cont.)
  • Increasing or decreasing
  • End behavior: What happens to f(x) as x grows large or small without bound?
  • General properties: Continuous? Differentiable? Power series for f?
  • Special properties: Symmetry, periodicity, connections to known functions
  • Models and Applications
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