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Piecewise and absolute Value Functions.pptx
- 1. Piecewise Functions Absolute ValueFuntions Prepared by: Ms. Shiela Salvador
- 2. Piecewise Defined Functions •In preparation for the definition of the absolute value function, it is extremely important to have a good grasp of the concept of a piecewise defined function. • Definition: A piecewise function is a function defined by two or more equations. Each “piece” of the function applies to a different part of its domain.
- 3. Evaluating Piecewise Functions Toevaluate a piecewise function, identify or figure out which piece applies to the given value of Consider the function Evaluate the following: 1. 6. 2. 7. 3. 8. 4. 9. 5. 10.
- 4. Domain of aPiecewise Function • To determine the domain of a piecewise function is simply to inspect where the given pieces are defined. Determine the domain of each piecewise function. Notice that is defined for all less than or equal to While is defined for all greater than . This means that the function is defined for all possible values of . Domain: set of real numbers 2. Domain:
- 5. Graphing Piecewise Functions •In graphing a piecewise function, we can follow the following steps: 1. Evaluate the endpoint(s) of function 2. Plot a few more points to give the shape 3. Repeat for each piece
- 6. Graphing Piecewise Functions Graphthe following piecewise functions and identify its domain and range. Example 1. At At Endpoint At – hole Graph is a horizontal line. Endpoint At – included Additional points: At At
- 7. Graphing Piecewise Functions •Graph of Domain: set of real numbers Range:
- 8. Graphing Piecewise Functions Example2. At At At Endpoint If , - hole Additional Points: If , If , Endpoints If , - hole If , - hole No need for additional points. Endpoint If , - included Additional Points: If , If ,
- 9. Graphing Piecewise Functions •Graph of Domain: Range: all real numbers
- 10. Graphing Piecewise Functions Example3. At At At Endpoint If , - included Additional Points: If , If , Endpoints If , - hole If , - included No need for additional points Parabola opening upward with vertex at y=-2x Endpoint If , - hole Additional Points: If ,
- 11. Graphing Piecewise Functions •Graph of Domain: all real numbers Range:
- 12. The Absolute ValueFunction • An absolute value function is a function that contains an absolute value expression. The can be written as a piecewise function: • Hence the graph is • Domain: set of real numbers • Range:
- 13. Graphing Absolute ValueFunctions The graph of has the following characteristics. • The graph has vertex and is symmetric in the line . • The graph is V-shaped. It opens up if and down if . • The graph is wider than the graph of if • The graph is narrower than the graph of if .
- 14. Graphing Absolute ValueFunctions Examples. Graph the following absolute value functions and determine its domain and range. Example 1. Vertex: Opening upward Symmetric at Domain: all real numbers Range:
- 15. Graphing Absolute ValueFunctions Example 2: Vertex: Opening downward Symmetric at Domain: all real numbers Range:
- 16. Graphing Absolute ValueFunctions Example 3. Vertex: Opening upward Symmetric at Domain: all real numbers Range:
- 17. Practice Exercises • Sketchthe graph of the following functions. Determine the domain and range. Show and indicated pertinent details.
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