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1.2 Functions and Graphs

Learn about functions, domains, ranges, and how to interpret graphs. Explore open, closed, and half-open intervals. Develop graph viewing skills and understand odd and even functions.

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1.2 Functions and Graphs

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  1. 1.2 Functions and Graphs Let’s talk variables… Until now, it’s been x’s and y’s so we’ll stick with that for a little.

  2. Functions A rule that assigns to each element in one set a unique element in another set is called a function.

  3. Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. In this definition, D is the domain of the function and R is a set containing the range.

  4. Function

  5. Example Functions

  6. Domains and Ranges

  7. Domains and Ranges • OUR INPUTS AND OUTPUTS CAN BE ALL REAL NUMBERS OR, JUST PARTS OF THE REAL NUMBER LINE. • WE USE THE FOLLOWING INTERVALS TO DENOTE CERTAIN DOMAIN AND RANGES: • OPEN, CLOSED OR HALF-OPEN, FINITE OR INFINITE. • THE ENDPOINTS OF AN INTERVAL MAKE UP THE INTERVAL’S BOUNDARY AND ARE CALLED BOUNDARY POINTS. • THE REMAINING POINTS MAKE UP THE INTERVAL’S INTERIOR AND ARE CALLED INTERIOR POINTS.

  8. Domains and Ranges • Closed intervals contain their boundary points. • Open intervals contain no boundary points

  9. Domains and Ranges

  10. Graph

  11. Example Finding Domains and Ranges [-10, 10] by [-5, 15]

  12. Viewing and Interpreting Graphs Graphing with a graphing calculator requires that you develop graph viewing skills. • Recognize that the graph is reasonable. • See all the important characteristics of the graph. • Interpret those characteristics. • Recognize grapher failure.

  13. Viewing and Interpreting Graphs Being able to recognize that a graph is reasonable comes with experience. You need to know the basic functions, their graphs, and how changes in their equations affect the graphs. Grapher failure occurs when the graph produced by a grapher is less than precise – or even incorrect – usually due to the limitations of the screen resolution of the grapher.

  14. Example Viewing and Interpreting Graphs

  15. Odd / Even • Odd functionf(-x) = -f(x) • Odd symmetric about origin • Even functionf(-x) = f(x) • Even symmetric about y-axis

  16. Example Even Functions and Odd Functions-Symmetry

  17. Example Even Functions and Odd Functions-Symmetry

  18. Example Graphing a Piecewise Defined Function

  19. Piecewise Defined Funtions

  20. Absolute Value Functions The function is even, and its graph is symmetric about the y-axis

  21. Composite Functions

  22. Example Composite Functions

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