Unit 3.2 Properties of Real Functions

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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
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.
• 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