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Understanding Functions: Key Concepts in Pre-Calculus

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32 diapositives · 497 Views · Aug 01, 2014

This guide covers vital concepts about functions in pre-calculus, starting with the definition of a function as a relation connecting each element of the domain to exactly one element of the range. It explains key terms such as independent and dependent variables, domain, range, and the different models for representing functions: numerical, graphical, and algebraic. The continuity of functions, types of discontinuities, and the concepts of increasing, decreasing, and bounded functions are also discussed. Understanding local maxima and minima, symmetry, asymptotes, and end behavior completes this comprehensive overview.

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  1. Pre Calculus Functions and Graphs

  2. Functions • A function is a relation where each element of the domain is paired with exactly one element of the range • independent variable - x • dependent variable - y • domain - set of all values taken by independent variable • range - set of all values taken by the dependent variable

  3. Mapping 3 -6 9 12 -1 5 0 -8 2

  4. Representing Functions • notation - f(x) • numerical model - table/list of ordered pairs, matching input (x) with output (y) • US Prison Polulation (thousands)

  5. graphical model - points on a graph; input (x) on horizontal axis … output (y) on vertical • algebraic model - an equation in two variables

  6. Vertical Line Test

  7. Finding the range • implied domain - set of all real numbers for which expression is defined • example: Find the range

  8. Continuity • http://www.calculus-help.com/tutorials • function is continuous if you can trace it with your pencil and not lift the pencil off the paper

  9. Discontinuities • point discontinuity • graph has a “hole” • called removable • example

  10. jump discontinuity - gap between functions is a piecewise function • example

  11. infinite discontinuity - there is a vertical asymptote somewhere on the graph • example

  12. Finding discontinuities • factor; find where function undefined • sub. each value back into original f(x) • results …

  13. Increasing - Decreasing Functions • function increasing on interval if, for any two points • decreasing on interval if • constant on interval if

  14. Example:

  15. Example:

  16. Boundedness of a Function

  17. Extremes of a Function • local maximum - of a function is a value f(c) that is greater than all y-values on some interval containing point c. • If f(c) is greater than all range values, then f(c) is called the absolute maximum

  18. local minimum - of a function is a value f(c) that is less than all y-values on some interval containing point c. • If f(c) is less than all range values, then f(c) is called the absolute minimum

  19. local maxima F I Absolute maximum B G A E J C K H Absolute minimum local minima D

  20. Example: Identify whether the function has any local maxima or minima

  21. Symmetry • graph looks same to left and right of some dividing line • can be shown graphically, numerically, and algebraically • graph: numerically

  22. algebraically • even function • symmetric about the y-axix • example

  23. odd function • symmetric about the origin • example

  24. Additional examples: even / odd

  25. Asymptotes • horizontal - any horizontal line the graph gets closer and closer to but not touch • vertical - any vertical line(s) the graph gets closer and closer to but not touch • Find vertical asymptote by setting denominator equal to zero and solving

  26. End Behavior • A function will ultimately behave as follows: • polynomial … term with the highest degree • rational function … f(x)/g(x) take highest degree in num. and highest degree in denom. and reduce those terms • example