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Unit 3.2 Properties of Real Functions - PowerPoint PPT Presentation

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

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

• Polynomial functions

• Rational functions

• Exponential functions

• Logarithmic functions

• Trigonometric functions (and their inverses)

• Sequences

• 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.

• 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.

• Type 1: called discrete real functions. Includes sequences.

• Type 2: called interval-based real functions. Includes the first 5 categories above.

• 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.) (p.91)

• 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