Session 11

1. Session 11 Agenda: • Questions from 5.4-5.6? • 6.1 – Piecewise-Defined Functions • 6.2 – Operations on Functions • 6.3 – One-to-One Functions and Inverses • Things to do before our next meeting.

2. Questions?

3. 6.1 – Piecewise-Defined Functions • A piecewise-defined function is one in which the function is defined separately on separate parts of its domain. For example: • Evaluate:

4. In the previous function, the domain is all real numbers. Why? • First, note that the three inequalities, x<-1, -1≤x<2, and x≥2 make up all possible values of x. • Secondly, notice that each function piece is defined for all the x-values in that piece’s domain. • Although is not defined for negative values of x, notice that we only use this piece of the function when x≥2.

5. Determine the domain of the following piecewise function.

6. Sketch graphs of the following piecewise functions.

8. A very important piecewise function is f(x)=|x|. Although it doesn’t appear to be a piecewise function, it is. • The domain of this function is all real numbers and the range is [0, ∞).

9. Consider the function • Evaluate: • Sketch a graph of the function using transformations.

10. To write the function as a piecewise-defined function, use the same general principles used for the basic absolute value function. which simplifies to: • Notice how these piecewise lines correspond with the graph you’ve drawn using transformations.

11. Consider the function • Find the x and y-intercepts. • Write f(x) as a piecewise-defined function and sketch a graph.

12. Consider the function • Write f(x) as a piecewise-defined function and sketch a graph.

13. Write the following function as a piecewise-defined function.[Hint: There will be three cases to consider.]

14. Sketch a graph of the function by first sketching a graph of and then reflecting the negative portion(s) of the graph across the x-axis.

15. 6.2 – Operations on Functions • Given functions f(x) and g(x), we can create new functions by adding, subtracting, multiplying, and dividing f and g. Domain: All values of x for which both f(x) and g(x) are defined. Domain: All values of x for which both f(x) and g(x) are defined, but g(x)≠0.

16. Consider the functions and .Find the following functions and their domains.

17. Function Composition • Two functions f(x) and g(x) can also be combined using function composition. • The domain of this composition is all values of x for which both g(x) AND f(g(x)) are defined. • Given f(x) and g(x), evaluate the indicated compositions.

18. Use the graph below to evaluate the indicated compositions.

19. Given f(x) and g(x) below, find the indicated functions and their domains.

20. Given f(x) and g(x) below, find the indicated functions and their domains.

22. Find functions f(x) and g(x) such that • Given f(x) and f(g(x)) below, what is g(x)?

23. 6.3 – One-to-One Functions and Inverses • A function is said to be one-to-one if every x-value in the domain corresponds to a differenty-value in the range. No two x-values can have the same function value. • Mathematically, a function f is one-to-one if whenever , it must be that . • Graphically, a function is one-to-one if it passes the Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects the graph at more than one point.

24. Determine whether the following graphs represent one-to-one functions or not.

25. Determine whether the following functions are one-to-one. If not, how can you restrict the domain to make it a one-to-one function?

27. Inverses • If is a one-to-one function, then it has an inverse function defined by: where y is any value in the range of f. • If f has domain A and range B, then its inverse function has domain B and range A. • Graphically, the graph of is a reflection of the graph of across the line y=x.

28. Suppose that f is a one-to-one function with the following function values: Determine the following inverse function values.

29. Consider the graph of the one-to-one function f below. Domain of f:____________ Range of f:_____________ • Evaluate: • Sketch a graph of the inverse function on the same set of axes. • Domain of :__________ • Range of :___________

30. By only sketching a graph of the function f below, determine the domain and range of the inverse of f. • Domain of :___________ • Range of :____________

31. If two functions are inverses of each other, then they satisfy the following properties: • In other words, a function and its inverse “undo” each other. • Verify that the following functions are inverses of each other.

32. To find the inverse of a one-to-one function, write y=f(x),interchange the variables x and y, and solve for y. The function attained is the inverse function. • Find the inverses of the following functions.

34. Find the inverse for the function below and sketch a graph of both the function and its inverse. State the domain and range of each. • Domain of f:____________ • Range of f:_____________ • Domain of :__________ • Range of :___________

35. Things to Do Before Next Meeting: • Work on Sections 6.1-6.3 until you get all green bars! • Write down any questions you have. • Continue working on mastering 5.4-5.6. After you have all green bars on 5.1-5.6, retake the Chapter 5 Test until you obtain at least 80%. • Make sure you have taken the Chapter 7 Test before our next meeting.