1. FUNCTIONS AND MODELS 1

2. 1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. FUNCTIONS AND MODELS

3. By applying certain transformations to the graph of a given function, we can obtain the graphs of certain related functions.  This will give us the ability to sketch the graphs of many functions quickly by hand.  It will also enable us to write equations for given graphs. TRANSFORMATIONS OF FUNCTIONS

4. If c is a positive number, then the graph of y = f(x) + c is just the graph of y = f(x) shifted c units upward. The graph of y = f(x - c) is just the graph of y = f(x) shifted c units to the right. TRANSLATIONS

5. If c > 1, then the graph of y = cf(x) is the graph of y = f(x) stretched by a factor of c in the vertical direction. The graph of y = f(cx), compress the graph of y = f(x) horizontally by a factor of c. The graph of y = -f(x), reflect the graph of y = f(x) about the x-axis. STRETCHING AND REFLECTING Figure 1.3.2, p. 38

6. The figure illustrates these stretching transformations when applied to the cosine function with c = 2. TRANSFORMATIONS Figure 1.3.3, p. 38

7. For instance, in order to get the graph of y = 2 cos x, we multiply the y-coordinate of each point on the graph of y = cos x by 2. TRANSFORMATIONS Figure 1.3.3, p. 38

8. This means that the graph of y = cos x gets stretched vertically by a factor of 2. TRANSFORMATIONS Figure 1.3.3, p. 38 Example

9. Given the graph of, use transformations to graph: a. b. c. d. e. Example 1 TRANSFORMATIONS Figure 1.3.4a, p. 39

10.  The graph of the square root function is shown in part (a).  by shifting 2 units downward.  by shifting 2 units to the right.  by reflecting about the x-axis.  by stretching vertically by a factor of 2.  by reflecting about the y-axis. Example 1 Solution: Figure 1.3.4, p. 39

11. Sketch the graph of the function f(x) = x 2 + 6x + 10.  Completing the square, we write the equation of the graph as: y = x 2 + 6x + 10 = (x + 3) 2 + 1. Example 2 TRANSFORMATIONS

12.  This means we obtain the desired graph by starting with the parabola y = x 2 and shifting 3 units to the left and then 1 unit upward. Example 2 Solution: Figure 1.3.5, p. 39

13. Sketch the graphs of the following functions. a. b. TRANSFORMATIONS Example 3

14. We obtain the graph of y = sin 2x from that of y = sin x by compressing horizontally by a factor of 2.  Thus, whereas the period of y = sin x is 2, the period of y = sin 2x is 2 /2 =. Example 3 a Solution: Figure 1.3.6, p. 39Figure 1.3.7, p. 39

15. To obtain the graph of y = 1 – sin x, we again start with y = sin x.  We reflect about the x-axis to get the graph of y = -sin x.  Then, we shift 1 unit upward to get y = 1 – sin x. Solution: Example 3 b Figure 1.3.7, p. 39

16. Another transformation of some interest is taking the absolute value of a function.  If y = |f(x)|, then, according to the definition of absolute value, y = f(x) when f(x) ≥ 0 and y = -f(x) when f(x) < 0.  The part of the graph that lies above the x-axis remains the same.  The part that lies below the x-axis is reflected about the x-axis. TRANSFORMATIONS

17. Sketch the graph of the function  First, we graph the parabola y = x 2 - 1 by shifting the parabola y = x 2 downward 1 unit.  We see the graph lies below the x-axis when -1 < x < 1. TRANSFORMATIONS Example 5 Figure 1.3.10a, p. 41

18.  So, we reflect that part of the graph about the x-axis to obtain the graph of y = |x 2 - 1|. Solution: Example 5 Figure 1.3.10a, p. 41Figure 1.3.10b, p. 41

19. Two functions f and g can be combined to form new functions f + g, f - g, fg, and in a manner similar to the way we add, subtract, multiply, and divide real numbers. COMBINATIONS OF FUNCTIONS

20. The sum and difference functions are defined by: (f + g)x = f(x) + g(x) (f – g)x = f(x) – g(x)  If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection.  This is because both f(x) and g(x) have to be defined. SUM AND DIFFERENCE

21. For example, the domain of is and the domain of is. So, the domain of is. SUM AND DIFFERENCE

22. Similarly, the product and quotient functions are defined by:  The domain of fg is.  However, we can’t divide by 0.  So, the domain of f/g is PRODUCT AND QUOTIENT

23. For instance, if f(x) = x 2 and g(x) = x - 1, then the domain of the rational function is, or PRODUCT AND QUOTIENT

24. There is another way of combining two functions : c omposition - the new function is composed of the two given functions f and g.  For example, suppose that and  Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.  We compute this by substitution: COMBINATIONS

25. Given two functions f and g, the composite function (also called the composition of f and g) is defined by: Definition COMPOSITION

26. The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.  In other words, is defined whenever both g(x) and f(g(x)) are defined. COMPOSITION

27. The domain of is the set of all x in the domain of g such that g(x) is in the domain of f. In other words, is defined whenever both g(x) and f(g(x)) are defined. COMPOSITION Figure 1.3.11, p. 41

28. If f(x) = x 2 and g(x) = x - 3, find the composite functions and.  We have: Example 6 COMPOSITION

29. You can see from Example 6 that, in general,.  Remember, the notation means that, first, the function g is applied and, then, f is applied.  In Example 6, is the function that first subtracts 3 and then squares; is the function that first squares and then subtracts 3. COMPOSITION Note

30. If and, find each function and its domain. a. b. c. d. Example 7 COMPOSITION

31.  The domain of is: Solution: Example 7 a

32.  For to be defined, we must have.  For to be defined, we must have, that is,, or.  Thus, we have.  So, the domain of is the closed interval [0, 4]. Example 7 b Solution:

33.  The domain of is. Solution: Example 7 c

34.  This expression is defined when both and.  The first inequality means.  The second is equivalent to, or, or.  Thus,, so the domain of is the closed interval [-2, 2]. Example 7 d Solution:

35. It is possible to take the composition of three or more functions.  For instance, the composite function is found by first applying h, then g, and then f as follows: COMPOSITION

36. Find if, and. Solution: Example 8 COMPOSITION

37. Given, find functions f, g, and h such that.  Since F(x) = [cos(x + 9)] 2, the formula for F states: First add 9, then take the cosine of the result, and finally square.  So, we let: Example 9 COMPOSITION

38.  Then, Solution: Example 9