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 Trigonometric functions (and their inverses) Sequences

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 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 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) 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.) 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