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Understanding Functions: Domain, Range, and Graphs

Learn about domain and range in functions, realistic values, vertical line test, increasing/decreasing functions, maximum/minimum values, even/odd functions, and how to find and interpret the domain and range of a function graphically. Homework included.

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Understanding Functions: Domain, Range, and Graphs

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  1. 1.3 Graphs of Functions Pre-Calculus

  2. Home on the Range • What kind of "range" are we talking about? • What does it have todo with "domain?" • Are domain and rangereally "good fun for the whole family?"

  3. Definitions • Domain: Is the set of all first coordinates (x-coordinates) from the ordered pairs. • Range: Is the set of all second coordinates (y-coordinates) from the ordered pairs.

  4. Domain • The domain is the set of all possible inputs into the function { 1, 2, 3, … } • The nature of some functions may mean restricting certain values as inputs

  5. Range { 9, 14, -4, 6, … } • The range would be all the possible resulting outputs • The nature of a function may restrict the possible output values

  6. Find the Domain and Range • Given the set of ordered pairs, {(2,3),(-1,0),(2,-5),(0,-3)} Domain Range

  7. Choosing Realistic Domains and Ranges • Consider a function used to model a real life situation • Let h(t) model the height of a ball as a function of time • What are realistic values for t and for height?

  8. Choosing Realistic Domains and Ranges • By itself, out of context, it is just a parabola that has the real numbers as domain anda limited range

  9. Find the Domain and Range of a Function • Find the domain of f(x) • Find f(-1) • f(2) • Find the range of f(x) *When viewing a graph of a function, realize that solid or open dots on the end of a graph mean that the graph doesn’t extend beyond those points. However, if the circles aren’t shown on the graph it may be assumed to extend to infinity.

  10. Domain and Range • Find the domain and range of

  11. Vertical Line Test • A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

  12. If a vertical line passes through a graph more than once, the graph is not the graph of a function. Hint: Pass a pencil across the graph held vertically to represent a vertical line. The pencil crosses the graph more than once. This is not a function because there are two y-values for the same x-value.

  13. The Ups and Downs • Think of a function as a roller coaster going from left to right • Uphill • Slope > 0 • Increasingfunction • Downhill • Slope < 0 • Decreasing function 19

  14. Increasing/DecreasingFunctions A function f is increasing on (a, b) if f (x1) <f (x2) whenever x1 < x2. A function f is decreasing on (a, b) if f (x1) >f (x2) whenever x1 < x2. Increasing Decreasing Increasing

  15. Example In the given graph of the function f(x), determine the interval(s) where the function is increasing, decreasing, or constant.

  16. Maximum and Minimum Values Absolute Maximum ( f (c1) f (x) for all x) Local Maximum ( f (c2) f (x) for all x in I • • |c2 |c1 I

  17. Maximum and Minimum Values Absolute Minimum ( f (c1) f(x) for all x) c2| |c1 • I I • Collectively, maximums and minimums are called extreme values. Local Minimum ( f (c2) f(x) for all x in I )

  18. Approximating a Relative Minimum • Use a calculator to approximate the relative minimum of the function given by

  19. Approximating Relative Minima and Maxima • Use a calculator to approximate the relative minimum and relative maximum of the function given by

  20. Temperature • During a 24-hour period, the temperature y (in degrees Fahrenheit) of a certain city can be approximated by the model where x represents the time of day, with x=0 corresponding to 6 am. Approximate the max and min temperatures during this 24-hour period.

  21. Piecewise Defined Functions • Sketch the graph of by hand.

  22. Even functions A function f is an even function if for all values of x in the domain of f. Example: is even because

  23. Odd functions A function f is an odd function if for all values of x in the domain of f. Example: is odd because

  24. Determine if the given functions are even or odd

  25. Graphs of Even and Odd functions The graph of an even function is symmetric with respect to the x-axis. The graph of an odd function is symmetric with respect to the origin.

  26. Determine if the function is even or odd?

  27. Determine if the function is even or odd?

  28. Determine if the function is even or odd?

  29. Homework • Page 38-41 2-8 even (graphical), 15-18 all, 19-29 odd, 32-36 even, 44, 48, 59-65 odd, 73-81 odd, 91

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