Understanding Functions: Key Concepts in Pre-Calculus

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

9. 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

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

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

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

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

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

18. Example:

19. Example:

20. Boundedness of a Function

23. 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

24. 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

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

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

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

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

29. odd function • symmetric about the origin • example

30. Additional examples: even / odd

31. 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

32. 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