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Pre Calculus

Pre Calculus. Functions and Graphs. 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

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Pre Calculus

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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

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