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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. Polynomial functions Rational functions Exponential functions Logarithmic functions

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Unit 3.2 Properties of Real Functions

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  1. Unit 3.2 Properties of Real Functions ‘Real function’ refers to a function whose domain and range are sets of real numbers.

  2. Categories of functions encountered in calculus and precalculus • Polynomial functions • Rational functions • Exponential functions • Logarithmic functions • Trigonometric functions (and their inverses) • Sequences

  3. 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.

  4. 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.

  5. The two types • Type 1: called discrete real functions. Includes sequences. • Type 2: called interval-based real functions. Includes the first 5 categories above.

  6. 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)

  7. 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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