Unit 3 2 properties of real 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. 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.) (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


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